UC-NRLF B 3 DTM 32fl TRANS ACTIONS OF THE •HILOSOPHICAL SOCIETY OF NEW ;3ouTH WALES, 1862-1863. SYDXEY: KD BY READING AN7D \ BEIIHiE iXEEET. TRANSACTION'S' PHILOSOPHICAL SOCIETY OF NEW SOUTH WALES. On fhe Vertebrated Animals of the Lower Murray and Darling, their habits, economy, and geographical distribution, by GERARD KREFFT. [Bead 10th September, 1862.] SOME of the observations embodied in this paper date back as far as the year 1852, but the greater number are the results of a nine months' sojourn on the Lower Murray and Darling, where, encamped in the neighbourhood of Gol Gol, I was enabled, with the assistance of Messrs. Williams and their devoted natives, to thoroughly investigate the fauna of that part of Australia. I cannot speak too highly of the valuable co-operation of these gentlemen, the same whom Mr. Landsborough met on his return journey at the Warrego pushing on towards the Gulf, and whom he designates " the best of Australian Bushmen." The Placental Mammals of this district are of course few. They belong to the Bats, (Nyctophilus, Scotophilus, Rhinolophus,) the Rodents, (Hydromys, Hapaloiis, MiisJ and the Carnivora, which are represented here, as in the rest of Australia, by the Dingo alone. CHEIROPTERA. The Bats of the Murray and Darling all belong to the family Vespertilionidae, as the large so-called " Vampire Bat " or " Flying Fox " (Pteropus) is not found in those regions. Owing to the superstitions of the natives, who look upon every Bat as a departed friend and relative, who, according to their ideas with regard to the transmigration of souls, has seen better days among themselves, has thrown spear and boomerang, and feasted upon Kangaroo, Wallaby, and Emeu, the number of Bats collected during my journey was very limited indeed. . 894053 2* V VERTfcAfTA OF THE LOWER MURRAY, '.? rvJviJi Sfct*{&n$ to 70 darker cross-bands, in some specimens the scales between the dark bands are an- teriorly edged with yellow, the two outer rows of scales yellowish, more or less clouded, but without any distinct spot in the centre of each scale as in H. superlus. Belly yellow, ventral plates frequently clouded or spotted with dark grey anteriorly, growing darker towards the tail ; the subcaudals, which are entire, being almost uniform blackish. The coloration of this snake varies considerably ; on the East Coast light-brown specimens are much more frequent than dark ones, whilst Western Australian snakes of this species are very dark-brown, and the cross-bands remarkably distinct. This reptile has been frequently alluded to by some authors as H. 8uperbu8y but I have always maintained that no continental species has ever been found with 15 rows of scales, and the vertical shield more than twice as long as broad ; the main characters by which the two snakes can easily^be distinguished. I am certain that more than 300 specimens have passed through my hands, and in not one instance did they answer to Dr. Gunther's description of H. superbus. I will give here the main points in which both Snakes differ : H. superbus. Scale* in 15 rows. Tail short, distinct from trunk. Head remarkably small, scarcely j distinct from trunk. Neck rather rounded. Scales of Head more or less elongate ; vertical, more than twice as long as broad. Coloration uniform brown, 2 outer rows of scales with reddish or yellow centre spot. Habitat Tasmania. Synonym Diamond Snake of the Tasmanians. H. curtus. 18 to 19 rows. Tail not distinct from trunk. Head very broad, as large again as H. Superbut, and distinct from neck. Neck very flat. Scales of Head very broad, in par- ticular the vertical, which with- out the anterior angle would form a square. Coloration brown banded, 2 outer rows of scales paler, or clouded with yellow and greyish. Habitat Australian continent. Synonyms Brown Banded Snake, N. 8. Wales ; Tiger Snake, Vic- toria. 54 OX THE SNAKES OF SVDNEY, I have had some correspondence with Dr. Albert Giinther regarding the habitat of the two Snakes, and I am glad to see the learned Doctor's statement in the Annals of Natural History for November, 1863, that " HoplocepUalus superbus proves to be a Tasmanian species." It would be interesting to know whether the Tasmanian Snake is able to inflate the skin of the neck when irritated, but judging from its small size this is not likely to be the case, and we must leave to Tasmaniau Naturalists the solution of this question. In the continental Snake the power to raise itself off the ground for half the length of the body, and to flatten out the neck like a Cobra, is well known, the Black Snake being the only other reptile which has been provided with the same power. A few words more and I have done with this, the most dangerous of all our Snakes. Its habitat is, I believe, the temperate part of Australia from East to West. I have taken it on the Murray, in South Australia and Victoria, and receive^ specimens from almost every part of New South Wales and from King George's Sound. The present species is not far removed from the Indian Cobra (Naja iripudians), and its bite is as deadly. A good sized dog bitten became para- lyzed within three minutes, and was dead in fifty minutes after- wards ; a goat died in thirty -five minutes ; another goat which escaped whilst experimented upon, was found dead in the street after a few hours ; a Dingo met the same fate in forty-eight minutes ; an Echidna (Echidna hysirix) lived six hours, and a Common Tortoise, an animal which will live a day with its head cut off, was dead in five hours after being bitten. Antidote vendors seeing the effect of the poison, never dared to peril their reputation in the attempt to save the animals so bitten ; I must mention, however, that in making these experiments* chance bites, where the snake makes a dart, bites, and retires, were out of the question, and I grant that under such conditions man or animal may recover ; but if the snake's head is applied to the lip or ear of some animal and the fangs well pressed into the wound, there is little hope of recovery. Let me also give a few words of advice to such men as go about exhibiting these reptiles, and showing their prowess by allowing themselves to be BY GERARD KREFFT. 55 bitten, professing that they possess an antidote against the poison ; generally speaking, these persons are more or less impostors ; they break off the fangs of the snake, but do not know- bow soon they are reproduced, and thus frequently fall victims to their ignorance. The Indian jugglers have more sense, and entirely remove the teeth, as most of the specimens of Naja tripndians prove which are received from India. The young of this snake, from 15 to 20 in number, are generally observed about the end of February ; they are then from 7 to 8 inches long, and subsist on small frogs, lizards, or insects. During the cold season this snake retires into the ground, as I have never met with half-grown or adult specimens under stones. PETROPYMOV. Krc/t. PETKODYMON CUCULLATUS. Red-bellied Si^e. Scales in 15 rows. Anal 1/1. Ventrals 187. Subcaudals 41,41. Purplish brown above, with a series of darker longitudinal lines along the upper part of the body, leaving a light elongate mark in the middle of each scale. Beneath yellow, bright red in adult specimens, each ventral plate clouded on the upper edge with purplish brown, much interrupted on the posterior part of the body. Divisional line of subcaudal plates marked in a similar manner, leaving the outer edges of the plates yellowish. Upper part of head purplish brown as far as the middle of posterior frontals, covering the vertical part of superciliaries, and reaching beyond the occipitals ; this elliptical spot is joined to the back by a narrow band of the same colour running along the median line of the neck. A light-greyish band encircles the dark-brown mark, divided by the narrow line by which this mark is joined to the back. Upper and lower labials dotted with brown spots. Body rounded, head rather flat, depressed ; tail 56 ON THE SNAKES OF SYDNEY, short, distinct from trunk, and ending in a conical spine or nail about a quarter of an inch long. Scales in 15 rows (not in 13, as mentioned by Dr. Giinther, whose description as Diemenia cucullaia, was taken from a very bad specimen) ; 6 upper labials, the third and fourth forming the lower edge of the orbit, the second labial not in contact with the posterior frontal ; rostral broad, low, very obtuse superiorly ; shields of the head regular, all more or less rounded posteriorly, and slightly imbricate, vertical twice as long as broad ; one anterior and two posterior oculars, one temporal in contact with both oculars, four or five scale-like temporals behind ; eye very small, pupil elliptical and erect. About 3 years ago — in 1860 — I captured a single individual of this species ; since then, owing to the exertions of friends in the country, specimens from Ash Island, Hunter River, Port Macquarie, the Clarence River, and other localities have been received, so that its geographical range has been ascertained for many hundred miles ^jpng the east coast. This snake is strictly nocturnal in its habits, sluggish and of gentle disposition, never offering to bite when handled, and though venomous, it is so in a very slight degree only, as has been proved by experi- ments ; its length seldom, if ever, exceeds 20 inches. Rocky and desolate places are frequented by it, and in such localities it is occasionally found under fiat stones during the cold season. VERMICELLA. Gray. VEBMICELLA ANNULATA The Ringed Vermicella. Scales in 15 rows. Ventrals 225. Anal bifid. Subcaudals 18/18. The following is Dr. Giinther's description : — "Body elongate, rounded, slightly compressed behind; tail very short; head BY GEEAED KREFFT. 57 moderate, not distinct from neck, similar to Elaps ; rostral shield very large, rounded, raised above the surface of snout ; occipitals rather narrow ; two posterior oculars ; anterior large, replacing the loreal together with the nasal ; hasal shield single, pierced in the centre by the small nostril ; six upper labials, third and fourth coming into the orbit ; one large temporal shield in contact with the upper posterior ocular, two smaller ones behind. Scales smooth, large, rather rounded behind, in fifteen rows. Anal and subcaudals bifid. Tail ending in an obtuse conical scale. Two small fangs in front of upper jaw, no other teeth behind ; palatine and mandibulary teeth equal in length. Crown of head and muzzle black ; a yellowish, in fresh specimens white, band across the posterior frontals, a second on the neck ; body and tail encircled by alternate black and white (in spirits) rings. Length of cleft of mouth y ; length of tail 1|" ; total length 28"." The ringed Yermicella, like all other nocturnal snakes, is very seldom met with, and apparently little known to the colonists. I often capture it during the cold season without taking any precaution whatever, as I know from experience that this gentle creature will never bite ; but even if it should do so, the wound would be small and of no danger whatever. I have never succeeded to make it bite of its own accord, but had to open its mouth forcibly if I wished to try an experiment. White, in his Voyage to New South Wales, gives a figure of this interest- ing snake, but little was known until a few years ago with respect to its geographical range. We find it as far south as Eden, Twofold Bay ; it occurs again in Western Australia, is tolerably common near Brisbane, and may probably be found much further north. Mr. William Taylor has lately presented a young specimen of this snake to the Museum, which was captured at the Culgoa River ; it is not unlikely that this species is found all over the continent from east to west. In its habits it is nocturnal, and closely allied to the genus Elaps, inhabiting South America ; in fact it bears, like our Bat- rachians, according to Giinther, a closer resemblance to the South American than to the Indian fauna. 58 ON THE SNAKES OF SYDNEY, ACANTHOPHIS. Daud. ACANTHOPHIS ANTARTICA. Wagl. , The Death Adder. Scales in 21 rows. Ventrals 127. Anal entire. Subcaudals 42. Head large, depressed, broad behind, regularly shielded, no loreal, 2 nasals, nostrils between ; 8 rows of dorsal scales, keeled to the root of the tail ; grey, sometimes salmon coloured above, minutely punctulated ; back and tail with about 4 or 5 white spots speckled with pink, lower lip flesh coloured (white or yel- lowish white in spirits), with a pale black dot in the centre of each scale ; beneath salmon coloured (yellow in spirits) ; tail dis- tinct from trunk, short, thin, and ending in a recurved soft spine. The colour of the Death Adder is subject to a good deal of variation, northern specimens from Rockhampton and Port Denison have the dark cross-bands of the back considerably smaller than those from the neighbourhood of Sydney, and the markings in the centre of the upper and lower labials and chin shields are of a pale greyish hue in the former. Specimens of a copper-red colour, as occasionally occur near Richmond, Rand- wick, and Long Bay, have seldom come under my notice from other parts of the continent. Its habits and economy are tolerably well known. It is fond of warmth and sunshine, frequents sandy localities, is sluggish in its movements, and does not jump backwards if going to bite. When irritated this snake flattens itself out generally in the form of an S, turning round to one side or the other with astonishing rapidity, but never jumping at its enemy. As regards the sup- posed venomous sting in the tail, I can assure everybody inte- rested in this matter that the caudal appendage is a mere orna- ment, quite soft, which nobody could run into his finger if he tried, and I am astonished that the fables which igno- rance has circulated in a former and darker age, have not been exposed long before this. In April or May they go into winter quarters, having during the summer months accumulated a sufficient quantity of fat, to BY GERAED KREFFT. 59 be under no farther necessity of catching frogs, grasshoppers, or field-mice during the next season. The burrow of some small rodent, or the hole furnished by a decayed root, is selected and taken possession of until the warm sunshine of spring recalls the sluggish reptile to fresh activity. I believe that the Death-adder is found in almost every part of Australia north of 36°. The Australian Museum is in possession of specimens from many parts of New South Wales and from various localities in Queensland. The British Museum received this snake from Port Essington and the north-west coast, and I have taken it myself on the Murray and Darling. Its length seldom exceeds 30 inches. A very large specimen measured 2 feet 2 J inches to the vent, and 4| inches to the tail ; total, 2 feet 7 inches ; around the body, 6 inches. 2. HTDROPHHXE, OR SEA SNAKES. PLATURUS. Latr. PLATURUS SCUTATUS. Laur. The Ringed Sea Snake. Scales (front part) 21 to 23 series. Ventrals from 213 to 241. Body subcylindrical, of moderate length, shields of the head subnormal in number and arrangement, nostrils lateral, in a single nasal shield, both nasals being separated from each other by a pair of anterior frontals. Scales imbricate, smooth, ventral shields well developed, tail with 2 series of subcaudals (Gthr.) Body covered with a series of black rings, 20 to 50 ; crown of the head black, the first and second black mark of the head and neck are joined below by a longitudinal band commencing from the chin ; snout and side of the head yellow, with a black band running through the eye (Gthr). This Snake is frequently thrown ashore after stormy weather near Manly Beach, Coogee Bay, Botany, and other localities. Its range is very extensive, and it is common in the Bay of Bengal, the China Seas, and on the Australian and New Zealand coast ; it lives on fishes, and is not much dreaded by the natives of the South Sea Islands who, I am told, handle this snake with impunity. 60 ON THE SNAKES OF SYDNEY. PELAMIS. Daud. Pelamis bicolor. Daud. The Black and Yellow Sea Snake. " Head long, with very long spatulate snout ; neck, rather stout ; body of moderate length ; nasal shields contiguous, longer than broad, pierced by the nostrils posteriorly ; only one pair of frontals ; scales not imbricate, not polished, tubercular or concave ; ventral shields none or very narrow ; lower jaw without notch in front ; 2 or 3 postorbitals ; neck surrounded by from 45 to 51 longitudinate series of scales : from 378 to 440 scales in a lateral longitudinal series between the angle of the mouth and the vent." (Gunther.) The coloration of this snake varies considerably ; the most prevailing colour is, the upper part of the head and the back uniform black, the sides and belly uniform brownish olive or yellow, the latter colour predominating just after the snake has shed its skin. Both the black and yellow colours are sharply denned. Tail with a series of black spots. This snake, which occasionally occurs on our shores, has a wide range, and appears to be as common on the Indian Ocean as it is here. The coast of New Zealand may be taken as its most southern limit. Dr. Gray, speaking about the Hydrides in the Brit. Mus. Cat. of Snakes, remarks " that they are true Sea- Snakes ; that they coil themselves up on the shore, living on sea- weeds, and lay their eggs on the shore." This observation is not correct if applied to the present species, as I have more than once taken gravid females with from four to six well-developed young of such a size as are sometimes met with swimming about, and apparently a few days old only. That they live on sea-weed is doubtful also, for though I have dissected almost every specimen which has come into my hands, I have found nothing but fishes or the remnants of such in the stomach. These are all the specimens of Snakes observed near Sydney ; and as the country has been well searched for more than five years, it will be difficult to discover new species. " Geometrical Researches " in four papers, comprising numerous new theorems and porisms, and complete solutions to celebrated problems, by MARTIN GARDINER, C.E. DEFINITIONS. 1. If A, B, 0, be any three points, then by " angle O • AB " we mean the angle formed by the revolution of a rigid straight line round O as centre from a position coincident with 0 A to a position coincident with OB, the method of movement being such as to sweep direct across the straight line AB from A to B. And according as the revolution is right-handed or left-handed we say the angle is of right formation or of left formation. 2. The " rotative " of a straight line A B in respect to a point O, is the method of rotatory movement of a rigid straight line round 0 as centre when the movement is such as to sweep direct across the straight line from A to B. And according as the revolution is right-handed or left-handed we say that A B is of right rotative or left rotative in respect to O. 3. By the rotative of a tangent drawn from a point to a curve, we mean its rotative in respect to the centre of the osculating circle at the point of contact of the tangent and curve. 4. By the term n'gon, we mean a figure composed of n con- nected portions of straight lines which we can conceive to be formed by n successive movements of one point. 5. The lines composing any n'gon are called sides • and the first point of the 1st side, and the final point of the nth side are called extremities. 6. A closed n'gon has its extremities coincident ; and an open n'gon has its extremities distinct. 7. When any n'gon is represented by means of the letters which indicate its first extremity and its various other angular points and last extremity written in successive order, we say it is of prescribed formation. 62 GEOMETRICAL RESEARCHES, 8. If 0 be a point, and A B a straight line : then by " rectare O'A B " we mean J 0 A • 0 B • sin (0 • A B) in which the lines O A, 0 B are regarded as of like signs, and in which account is taken of the formation of the angle 0 • A B. 9. By the " rectare " of any closed n'gon AX A2 . . . . AW AX we mean the sum of the n products. J O A1 . O ' \ ' sin (0 • AJ A2) i OA2-OA3-sm(0 -A2A3) I OAn- OAj-siii (0- A^Aj) in which 0 is any fixed point in its plane, and in which all the straight lines issuing from 0 are regarded as of like signs Ifgp0 The rectare of a closed n'gon of prescribed formation is constant in sign and magnitude. 10. If S represents any straight line or plane, and that p represents any point ; then p, S represents the length of the perpendicular from^> on S. I think it proper to mention, that early in 1862 I forwarded enunciations of the theorems in papers 2 and 3, and some of those of paper 4 to the President of the Queen's College Galway, and to Professor Chasles. I sent also enunciations of the princi- pal porisms and theorems in papers 1 and 2 to Sir James Cockle, the Chief Justice of Queensland, immediately after his arrival in the Colony. BY MARTIN GARDINER, C.E. (53 PAPER No. 1. Researches concerning figures peculiarly derived from other figures by MARTIN GARDINER, C.E. [Read 9th July, 1862.] Given a closed n'gon AX A2 ............. AW AX , to find the locus of a point O, such, that if we join the successive points Bj , B2 , ..... BM , BX , of the feet of perpendiculars from it on the respective sides A^A^, A^ A^ , . . . . An AI , of the given n'gon, then will the rectare of the new n'gon BX B2 ---- Bn BX , thus derived (which we will call the derived figure) be of a given magnitude, £. PROCESS OF INVESTIGATION. Let alt a2 ,...., an , be the centres of the circles circum- scribing the quadrilaterals Now, paying attention to the formations of magnitudes, and looking on the quadrilateral 0 Bn AX BX , we at once perceive that J rectare (OB^ +0^6^ = rectare (OB^) -rectare (a^B^Bj). And from the quadrilateral 0 BX A2 B2 , we have f rectare (CKB^ + O'AgBg) = rectare (OB^)- rectare (a^BJ. And in the like manner we get the following equations from the other n — 2 quadrilaterals : — i rectare (CKB^ A, + OAnBn) = rectare (OB^BJ- rectare (an But if w-j, w2, m3, represent the feet of perpendiculars from the centre of the circumscribing circle on the sides AI A , A^ A3 y A3 Aj , of the triangle and that we put S" to represent the rectare of the triangle ml «^2 m3 m^ , then — £' = — 4J£", and therefore we have - R2 66 GEOMETRICAL RESEARCHES, 3. If the figure A^... -A^ is a complete regular polygon of n sides ; than the centre M of the locus of O is coincident with the centre of the circle which circumscribes the polygon. And if we represent the radius of the circumscribing circle by R, it is evident we can write the equation of the locus in the form n . sin ( - revolution) . p2 + n . sin (- revolution) R2 = 85* — - 4^' in which the involved angle is of like formation with ^". From this we at once see that when n = 2, we must have 85* = 45" ; and /. as % = zero, so also must 2 = zero. When n = 4, then we must have 8^ == 4^', or %' = 22 ; and the problem is porismatic. 4. Again, since sin (| revolution) = i chord (^ revolution,) and that ^ chord (— revolution ) = perimeter of polygon of j sides ; therefore it is evident that n . sin (|- revolution) is equal to twice the perimeter of a polygon of -| sides. And /. it is evident that when AI A2 . . . A^ AI is a regular polygon, and n = infinity, we have n- sin (-• .re volution) = 4 TT = 4 (3 1416) and 4 TT . p2 + 4 TT R2 = 8£ - 4^.' 477.^ = 82 - 8^" 7T.p2= 2 (S ~ 5') And, putting 2" to represent the rectare of the circle which con- stitutes the locus of O, we can write the last equation in the form 5" = 22 - 22' When we suppose p = H, then the derived figure of the circle AI A2 . . . A^ AI is a cardoid whose rectare is represented by 5*. and therefore, as in this case, the point O has the circle A A . . . . AM A as locus, we have BY MARTIN GARDINER, C.E. 67 5. It is evident we can investigate in like manner (and that the locus of 0 will be a circle) when, instead of one closed n'gon> we are given any number of closed n'gons, and that the sum of the products of the rectares of their derived figures and given numbers of known signs, is to be of a given magnitude. And it is also evident the locus of 0 is a circle when the sum of the products of the rectares of some of the derived figures and given numbers, has a given ratio to the sum of the products of given numbers and the rectares of the remaining derived figures. 6. Again (owing to the nature of the investigation, and to our knowledge of the relative properties of approximate figures) it is evident the principle of continuity justifies the extension of our results to the more comprehensive propositions in which given straight lines are replaced by curves of any kind whatever in a plane. And (remembering that the curvatures at points in curves are proportional to the angles between tangents at the extremities of equally long elements} we may announce the following important porisms : — PORISM I. Given any lot of closed figures of prescribed formations in a plane, then will the locus of a point o be a determinable circle when the sum of the products of given numbers and the rectares of the derived figures of the lot in respect to this point is of a given or determinate magnitude %. And, for all values of %, the centre of the determin- ate circle (whose circumference is the locus of o) is a fixed point, with which the locus of o is coincident when 2 is a minimum, and which point is the mean centre of the curvatures when the given figures are closed curves and the given numbers all equal. PORISM II. Given a lot of closed figures of prescribed formations, and given also a second lot of closed figures of prescribed formations ; then will the locus of a point o be a determinable circle when the sum of the products of given numbers and the rectares of the derived figures of the first lot in respect to the point, has a given or determinable 68 GEOMETRICAL RESEARCHES, invariable ratio to the sum of the products of other given numbers and the rectares of the derived figures of the second lot in respect to this same point. 7. These porisms give immediate intimation of numerous interesting theorems, of which the two following are ex- amples : — THEOREM I. The rectare of the derived figure of any conic in respect to any point in the circumference of a circle having the line joining the foci as diameter, is equal the rectare of the circle. THEOREM II. If any number of conies have a common focus ; then will the locus of a point o be a determinable circle, passing through this focus, when the sum of the rectares of the derived figures of the conies in respect to the point o is equal to the sum of the like rectares of the circles having the transverse axes of the conies as diameters. 8. In respect to the general problem, it is evident that when the given data is wholly or partly curve, the exact locus of 0 cannot be (unless in some particular cases) obtained without the aid of the infinitesimal calculus. It is also obvious that in cases in which some points of the given data are at infinity, the co-ordinate methods will afford the best means of actual solution, though of course the principle of continuity justifies us in predicting the nature of the locus, even when the manner of approximating to its position (as indicated in our general investigation) may not be intelligible to our limited understanding. However, to clear all doubt on this point, we can easily find the equation of the locus of O without paying attention to the rectare of the given figure AIAa....AnAl. This may be done in various ways, but the following is sufficient : — BY MARTIN GARDINER, C.E. 69 Assuming rectangular axes of reference, let G1 x + Hx y + KL = 0 H2 y + ^ = 0 Gn * + Hn ?/ -f Kn = 0, be the equations of the n successive straight lines A^ K^ Ag A ...... AW AI? taken in order. Then putting x', y't to represent the co-ordinates of any one of the positions of O, it is evident we can express the equations of perpendiculars from this point to the lines represented by the above equations, and that we can find the co-ordinates of the feet B , B2, . . . B , of these perpendiculars in terms of x', y', and known quantities. Hence it is obvious that if we indicate the co-ordinates of B , B2, ..... BW, by (*!> 2/1) > Ov ya)» • • • • (*V ^» ^ fche eciuatio11 then by substituting the values of xl y^ x^, y2, &c., implicating x'j y', and known quantities, we will have the equation of the Circle which is the locus of O. And by expressing the equations of the sides of the given figure AX A2 . . . . AM Aj in terms of the co-ordinates of the points AI? A2, . . . A^, we can arrive at the theorems already found, but not so obviously as by the method already exposed. 9. Theorems pertaining to all kinds of plane figures are very limited in number, owing no doubt to the bias for investigations concerning peculiar forms which the minds of geometers suffer in learning the science of geometry. When theorems of such a general nature are discovered they should not be passed unnoticed in elementary class-books ; for they show to us that in geometry (as in nature) we may have forms of the most irregular character adapted to fulfil definite 70 GEOMETRICAL RESEARCHES, relations in as complete a manner as forms definable by words or equations. In addition to the theorems given in the preceding investiga- tion, I would direct attention to the following general theorem. THEOREM. If a rigid tangent of fixed length perform (as tangent) a 'movement round any given complete curvilinear plane figure so that the point of contact continues at the extremity of the tangent ; then if m denote the number of right loops, and n the number of left loops which constitute the given figure, the rectare of the track of the out- ward extremity of the tangent will differ from the rectare of the given figure by (m — n) times the area of a circle having a radius equal in length to the tangent. NOTE. — The porisms evolved in this paper cannot fail to be interesting to geometers who apply themselves to questions in Speculative Astronomy. PAPER No. 2. Researches concerning n'gons inscribed in other n'gons by MARTIN GARDINER, C.E . [Read 9th July, 1862.] Given n straight lines LL, L2, ...... L^ and the like number of points ov o2, on in a plane ; to describe a closed n'gon pl p2 pn p^ which will have its points pv p2, .... jpw? on the straight lines Lp L2, LM taken in order, and its successive sides pl p2, p2 p3, PnP\ Pagsing in order through the n respective points ov o , .... o . BY MARTIN GARDINER, C.E. 71 PROCESS OF INVESTIGATION. 1. If we assume any three points alt b^ c^ in the line L , and that we draw the straight lines «x o^ 6X o^ c^ ol to cut L2 in the points «2, b2, c2 ; then Similarly, if we draw a o. 60 o0, ca o,, to cut L0 in points 8 « * * B Z o * a*> b9. c9 ; and that we draw a o_, 6_ o,, CQ o,, to cut L in OOO OOOoOd ^ «4, 6 , c^ ; and that we proceed thus until an on, bn o , c on, cut LX in an 1? 6n 1? CM x ; then evidently we have the following relations (each one of which is similar to the above) : — n+1 n+l Cn+l And from these n equations we at once derive the equation : — — ^- : ~— = — — : *+1 : (!)• n+l n+l From this we learn that pl is known (by the problem of " determinate section " of Apollonius, or because it is a double point of known homographic divisions on LX) ; and therefore also Moreover, we learn that in the non-porismatic state of the data, there are two and but two positions for pv both real or both imaginary. 72 GEOMETRICAL RESEARCHES, 2. When the data is so related that the points an+l, &n+1, «w+1 fall on the respective points a , Z^ , Cj ; then evidently the problem is porismatic ; for equation (1) assumes the form which holds when p^ has any position whatever in LX . Hence we may announce the following theorem : — THEOREM I. If there be n straight lines and n points in a plane, and that any three closed logons can be described, such that each one of them has its n angular points in the n respective lines, and its n sides passing in order through the n points, then ivill any point in any of the fixed lines be an answerable position for the prescribed angular point (resting on that line) of a closed u'gon fulfilling the like conditions with the three others. 3. Theorem 1 will enable us to arrive at some interesting porisms and theorems. Firstly. — Suppose we were given all the data but the two points on_1 and on, and that it is required to find such positions for these points as will render the problem porismatic. Here our object is to form 3 closed ft'gons ^ a ... a av b ^2 • ' ' ' ^n ^i' °i C2 * • • • cn c'i> (su^Ject to ^6 imposed conditions as respects the given data) whose sides an__J a^ bn_l bn, cn_l c^ will pass through one point, and whose sides an a , b b , c c , will pass though another point : for these two points would evidently be answerable positions for c^__ and on. If we take b1 in the intersection of Lj and LW, we can find the corresponding point b on L . If we take a . in the intersection of L and L n — 1 n — 1 n — 1 » — 1 nt we can find the corresponding point a1 on Lr And if we take GI anywhere in LA, we can find the corresponding point cn_1 on the BY MARTIN GARDINER, C.E. 73 line L .. Now if we take any point in ^ bn_l as a position for OM_I, and that we draw o cn__l to cnt L^ in c^ ; and then draw c c, to cut a a in o : it is evident o . and o so determined n 1 1 n — 1 n * — i *> are answerable positions : for we have the three closed w'gons, cn ci Dialling the conditions. Hence in the general states of the data, we have the following porism : — PORISM I. If a closed n'gon have its n angular points on n given straight lines, and have its first n— 2 sides passing through n — 2 given points; then its n — 1th and upsides will cut two determinate straight lines XX, TY in points on_1 and on, such that if we look on these points a$ fixed, we can " deform" the n'gon so that its angular points will move along the n given straight lines, and its sides continue through the n fixed points. In respect to this porism it may be proper to observe that when the two lines L , and L are parallels (and therefore a n— 1 n n—1 at infinity) and that a^ is at infinity on 1^, then the line a^ a^_1 is at infinity ; and the point on where GI cn cuts al (^n_1 is at in- finity ; and therefore the last sides of the w'gons (the sides through o ) must be all parallels to each other. And if L , L 0 72 A ft fl— 1* and L be parallels, and that a{ is at infinity, then bl bn_1 and a an_l are at infinity ; and it is evident we can assume c^ any- where on L^, and that cn_l c-n and cn cl will continue through on_^l and OM at infinity. 4. We arrived at porism 1 under the hypotheses that LX, '^n_v and L do not intersect each other in one point ; and therefore it is necessary to inquire into the nature of the relations when these three lines pass through one point. 74 GEOMETRICAL RESEARCHES, When such is the case, it is evident the line bl 6n_1 (or XX) becomes co-incident with LM—I, and that the line a1 a-n_l (or YY) becomes co-incident with 1^). It is also evident o%_1 can be taken anywhere in Ln_1 ; and that on can be taken anywhere in Lr It is also evident the nth angular points of the %'gons are all co-incident in the common point of intersection of the lines L1? LW_I» and LB. Moreover, we arrive at this under the hypotheses that all the given straight lines do not pass through one point. Hence we infer the following theorem : — THEOREM 2. Given n straight lines L,, L2, L , of which the first, the n — 1th, and the nth pass through one point, — the rest not all passing through this point ;. and given likewise n — 2 point ov o2, °w_2> °f a serws of n points : if positions for the n — 1th and nth points o%_1 and on of the series be such that any point whatever in the line L is an answerable position for the first angular point of a closed tig on p1 p2 .... pM p1? having its n angular points p p ... PM on the n respective lines Lp L2, .... LM, and its n sides p p p2 p3, . . . . pn p1? passing in order through the n respective points o,, o0, . . . . o , then will the points o , and o be situated in the 1 2 n7 •*• n — 1 n given straight lines LW_I, L/t, each in each respectively. 5. When all the given straight lines L , L2, . . . . L , intersect in one point q, then it is obvious that the infinitely small %'gons a\ a2 " • an—i an ai> &i ^2 ' ' ' ' ^n ^i are not Distinct, and that * i ^M_i an(i ai aw_! (XX and YY) are not determined in position. In this case we may evidently assume o^_1 anywhere in the plane (because a^ <^n_l is not restricted in position) and find corres- ponding answerable positions for o in the intersection of the sides c1 cn, dl dn, of any other two closed w'gons GI c2 . . , . on_l BY MARTIN GARDINER, C.E. 75 c c,cJdc.....d , d cL. fulfilling the conditions (because the Tl 1 L £t n — 1 ft A * three ?&'gons including these two and the n'gon a^ a . . . & n\ fulfil the conditions). Moreover it is evident that if we draw any straight line XX through q ; then for all points o assumed in XX3 the corresponding points o will be on a deter- minable straight line passing through q. Hence we have the following porism :— PORISM 2. If all the angular points of a closed n'gon move on n given straight lines meeting in a point q, and that all its sides but the nth pass through determinable fixed points ; then if the point through which the n — 1th side passes be situated on a known straight line passing through q, so will the point through which the nth side passes be on a determinable straight line passing through q. (See Mulcahy's " Modern Geometry," page 77.) 6. If we have the n— 2 given points ov ov .... on_2, in direc- tum with the intersection of LX and Ln ; then (no matter how general otherwise the given lines may be) it is evident the straight line X X will pass through these n— 2 given points ; and therefore on_1 will be in directum with the n — 2 given points. Hence we may announce the following porism : — PORISM 3. If all the angular points of a closed iCgon move on given straight lines, and all its sides, except one, pass through given points which lie in a straight line passing through the intersection of the lines on which the extremities of the free side move, then this side also passes through a fixed determinable point. (See Mulcahy's " Modern Geometry," page 77.) 7. When the given points o1, oa, 2 ... , as the point through which L must pass, then there corres- ponds another point in the same line through which L must pass. And we have the following porism : — PORISM 6. If all the sides of a closed n'gon pass through given points which lie in one straight line, and that all its angular points except one move on given straight lines : then will the locus of the free angle be a determinable straight line (see Mulcahy's Geometry, page 75). 11. Porisms 5 and 0 can be easily derived from those which precede them by the usual method of reciprocation ; and other particular theorems and porisms can be deduced from these, &c. However, I will not enter more into details in the present paper, as my chief object is to get at the more general relations. 12. Thirdly. — Given all the data but the point o and the line L to find positions for these which will render the problem poris- matic. Let a1 a2 an_l a^ and &x &2 bn_: b^ be the two known closed (n — l)'gons, having their sides passing in the prescribed manner through the n — 1 given points, and their angular points on the n — 1 given straight lines. If we put b for the point in which the straight line &w_1 b1 on_l cuts L/n ; then evidently (pn having any position not in «w_1 b^ 0f|_1) we must have bn coincident with 6 , or, in other words, we must have L passing through 6 , the first angular point of one of the BY MARTIN GARDESTER, C.E. 79 known closed (11 — l)'gons. And, putting an the point in which the straight line an_1 a1 on_i cuts Ln, it is evident that for the w'gon al a2 . . . o^ an a^ we must have any answerable point on situated on the straight line a an which is the last side of n — 1 1 n — 1 the other known closed (?i — l)'gon. And if Cj ca . . . . c e1 be any closed rc'gon, having its first n — 1 sides passing through the n — 1 given points, and having its first n — 1 angular points on the n — 1 given lines; it is evident that by drawing a straight line from b through c , and by producing c^ c^ to cut the line on_} ^n_l a1 ; then will this point of intersection and the line b± cn be answerable positions for on and L^ : (because we have the three closed rz/'gons al a .... a a1? b1 b^ . . . bn b^ °i C2 '" Cn ci' ^filling the conditions). Hence the following porism : — PORISM 7. If a closed n'gon have its first n — 1 angular points on n — 1 given straight lines, and its first n — 1 sides passing through n — 1 given points : then tv:o straight lines and a point in each of tliem can be found, such that if from either one of these determined points (in a determined line) we draw a line Ln through the nth angular point of the n'gon, and chat we produce the nth side of the n'gon to cut the other of the two determined lines in a point on, and that we regard th& line L and point on as fixed : we can deform, the n'gon so that its n sides v:ill continue through the n points composed of the n — 1 given ones and, the determined one o , and its angular points move along the n straight lines composed of the n — 1 given ones and the determined one LB. 13. Be-considering the problem, it is evident we can use the following system of equations : — 80 GEOMETRICAL RESEARCHES, Vi And from these we at once obtain the equation : — b2°l bn+l °n bo a , o n n n+l n In the porismatic states of the data it is obvious the second side of this equation must = 1, since a l & & 1 must be coincidents with c^ & & . 14. We may also remark that if a^ a^ »w ax & ll &2 .... & 6 are two closed %'gons, and that o^ o , .... o the intersections of their pairs of sides (of their first sides ; of their second sides, &c.) are in one straight line ; then, drawing straight lines through «r and 6p through a^ and &a, &c., to cut the line ol o2 on ; it is evident the line QI o2 .... on has coincident with it a third closed %'gon inscribed in the n lines &, &i> ao b^ &c. And therefore we must have : — 11 92 °n ai °n or \ ( \ °| ' \ °2 ........ 61 Q»\ = \6l °r&2°2 ........ bn °J ' And as the third or single-line closed n'gon can be substituted instead of either of these, we must have each of the two factors of the first side = 1. BY MARTIN GARDINER, C.E. 81 15. The problem can be investigated in the following manner : — Looking on the four successive sides pl p_^ p9 ps, p3 p^ p^ p& let us see whether we could replace them by a less number of sides passing through determinable points, and having their inter- sections on determinable straight lines, and their extremities inp1 and p.. Let ij be the point of intersection of the straight lines o og and o 03. Then since o , o2, i , are in one straight line, it follows that the intersection TI of the straight lines oj p2 and ^ p3 is in a known straight line Rr And since o3, o4, i^ are in one straight line, it follows that the intersection sl of the straight lines o p± and ^ p3 is in a known straight line S . Hence it is evident the solution of the problem is reduced to that of describing a closed (n — l)'gon pl rlslP5P6 ••••• PnPi, whose sides pl rv r{ s^ sl p., p^p& pn pi pass through the n — 1 known points o^ i^ o^ og, on, and whose angular points jp1? r1? 8V p5, . . . . p will rest on the n — 1 known straight lines LX, B , S1? L5, .... L^. Similarly, by proceeding with the first four sides of this closed (n — l)'gon as with those of the fz'gon, we can reduce the solution to that of describing a closed (n — 2)'gon pl r2 s2 pQ . . . . pn pl whose sides pass in order through n — 2 known points o^ i.^ o5 . . . . on, and whose angular points p^ r2, s2, p6 . . . . pn rest on the n — 2 known straight lines Lj, R2, S2, L6, . . . . Ln. And thus, step by step, we can proceed until we make the solution of the problem depend on that of describing a triangle Pl V3 S«-3^1 WhOSG SideS Pl rn-3' Tn-3 Sn-V Sn_3 PI> wlU P338 through known points 01? * _g, o , and whose angular points Pv rn-3 sn-3? w^^ rest on ^nown straight lines Lp Bn_3, Sn_3> 82 GEOMETRICAL RESEAECHES, 16. To arrive at porismatic relations of the data let us consider the question when we arrive at that point in the investigation where we have reduced the solution to the forming of a quadri- lateral pl rn_^ sn_4 pn p1 whose sides pass in order through the known points o^ \_^ °n_^ °n-> and whose angular points rest in order on the known straight lines Lp Rn_4, &„_# ^V It is evident our method of investigation leaves the point o and the lines L^5 Lp unimplicated ; and therefore it follows that if we give such positions to these as will render the forming of the quadri- lateral porismatic, then will the problem of the forming of the w'gon p1 p2 • •" PnP1 be porismatic. Now, if we assume any two positions for LI and Ln which pass through q the point of inter- section of R , and S , then (by a well-known porism) we can find a position for o which will render the problem of the form- ing of the quadrilateral porismatic. Hence we may announce the following porism : — PORISM 8. If a closed rfgon have all its angular points Taut the 1st and n111 resting on given straight lines, and have all its sides 'but the IIth passing through given points : a point q can be found, such that if we look on straight lines from it through the 1st and n^ angular points as fixed, and that we deform the n'gon so that its n angular points will move along the n straight lines composed of the n — 2 given ones and the two described ones, and so that its first n — 1 sides continue through the n — 1 given points, then will the n^or last side of the n'gon continue through a fixed determin- alle 17. The problem can be investigated in the following manner : Suppose we draw the straight line Qx containing the inter- section of LI and L2, and that of L3 and L4. And let ql be the point of intersection ofp2 p3 and Q^. BY MAKTIN GARDINER, C.E. 83 Then since LX, L2, and Ql pass through one point, it follows (by a well-known porism) that p ql cuts the line ol o in a known point /-j. And since L3, L , and Qj pass through one point, it follows that p q cuts the line o., o in a known point « . Hence evidently the solution of the problem is dependent on that of forming a closed (n — l)'gon p n p^ .... pnp1 whose successive sides pass through the n — 1 known points r , ^, o4 . . . . on, and whose angular points taken in order rest on n — 1 known straight lines L^, Q1? L4, .... Ln. And, proceeding with this closed (n — l)'gon as we have done with the closed w'gon, we can reduce the solution of the problem to the forming of a closed (n — 2)'gon plq2p . .. . Pnp^ whose sides pass in order through n — 2 known points rg (in line s1 r1), s2 (in line sl o^), o_, og, ---- o^ and whose angular points rest in order on n — 2 known straight lines LI} Q2, Lgj And thus, step by step, we can proceed until we make the solution depend on that of forming a triangle p1 qn_^ pn pv whose sides pass in order through three known points ^n_3, sn_3> °n> an(^ whose angular points rest in order on three known straight lines 18. To arrive at porismatic relations, we will (as in last method) consider the question at that point in the investigation where we have reduced the solution of the problem to the forming of the quadrilateral p± gn_4 pn_l pn p^ having its angular points, taken in order, on known straight lines LI} Qn—4> ^Jn_1> ^ and its successive sides passing through four known points r^ ^ s ,o , o . For, as the method leaves the points 0,0 , and n — 4 n — 1 n n « — -1 the line L unimplicated, it follows that if we give these such 84 GEOMETRICAL RESEARCHES, positions as will render the quadrilateral porismatic, then will the problem of the forming of the complete %'gon be porismatic. But it is evident (from a well known porism) that if we assume any two points in the straight line rn_4 *n_4 as positions for o , and o , then we can find a position for L so as to render »— 1 n' n the problem of the forming of the quadrilateral porismatic. Hence we may announce the following porism : — PORISM 9. If a closed n'gon have its first n — 2 sides passing through n — 2 given points, and have its first n — 1 angular points resting on n — 1 given straight lines : a straight line can be found, such that if we look on the two points in which it is cut by the two last sides of the j^gon as fixed, and that we deform the n'gon so that its sides will continue tlirougli the n points composed of the n — 2 given ones and the two determined ones, and that its first n — 1 angular points move on the n — 1 given straight lines, then will the locus of the nth angular point of the n'gon be a determinate straight line. This porism is evidently derivable from Porism 8 by reciprocation : 19. It is evident from the properties of homographic pencils and divisions (see Chasles' " Geometric Superieure "), that we can solve the more extended problem, in which all or any number of the entities o1, oa, .... on may be replaced by conies to be touched by sides of the closed w'gon, provided these conies touch the respective pairs of given straight lines on which the extremities of the touching sides are to rest. And all the data but the conic on being given, the method of finding this conic so as to render the problem porismatic is evident. And if instead of requiring all the angular points of the closed %'gons to rest on straight lines, we were to have all or any number of them rest in given circles, or other given conies passing through the pairs of given points through which the sides of the w'gon forming such angular points pass, then also we can solve. BY MARTIN GARDINER, C.E. 85 The numerous problems which may be formed by di- versifying the data give rise to porisms which may be easily evolved. I will not enter on their investigation in this paper ; but the following porism, comprising a multitude of particular cases, can be easily deduced : — PORISM 10. If there be a closed n'gon, having its 1st and n — 1th angular points resting on given straight lines LI? Ln_1} and that the nature of the conditions imposed on the n — 2 first sides and angles be such that by forming the open (n — %ygo-ns according to these conditions, we shall have the given straight lines Lj and Ln_1, divided homographically by their extremities : two straight lines XX, YY, can (generally) be found, such that if we look on the points o and o in which they are cut by the n — 1th and n* sides of the closed tigon as fixed points, and that we " deform " this u'gon so that its sides and angles continue subject to the imposed con- ditions, and that its n — 1th and n* sides continue through, the deter- mined points o and o , then ivill the nth angular point of the n'gon describe a determinable circle passing through o , and o „ tt— 1 (see Porism X. in Transactions of the Royal Society of Victoria for 1859). 20. From the well-known properties of three pairs of points in one straight line, which are in involution (see " Geometrie Superieure " ) we infer the following theorem : — THEOREM 4. If we can form one closed 2 n'gon whose first n sides and whose second n sides pass successively through n fixed points taken in prescribed order, and whose first n angular points rest in succession on n given straight lines taken in prescribed order : then any point whatever in any of the lines is an answerable position for an angular point of a like closed 2 n'gon ; or which amounts to the same — we can deform the 2 rigon subject to the imposed conditions, so that its angular points will move along the n straight lines. 86 GEOMETEICAL RESEARCHES, 21. The problem solved at the commencement of this paper may be regarded as strictly analogous to the following one : — " Given a system of n straight lines LI? L*2, .... Ln in space, and given also a second system of n straight lines K1? K2, .... K^ in space ; through the lines of the second system taken in order to draw n planes forming a closed planes n'gon whose n angular joints will rest on the n lines of the first system taken in order." To those who understand the homographic theory (and possess the ability to conjure up figures in the air) the method of solution is obvious ; and the following theorems and porisms are evident consequences. THEOREM 5. If we can form three closed planes w'gons, such that the n planes of each contain n fixed straight lines in space (each plane containing a certain line), and the n joints rest on n other fixed straight lines in space ; then will any point whatever in any straight line of the second system of n lines be an answerable position for an angular point of a closed planes n'gon fulfilling the like conditions. THEOREM 6. If we can form one closed planes 2 w'gon, whose first n planes and whose second n planes contain n fixed straight lines (in space) taken in prescribed order, and whose first n angular joints and second n angular joints rest on n other fixed straight lines taken in prescribed order ; then we can deform this planes 2 Ti'gon so that its sides will contain the straight lines of the first system, and its angular joints move along the straight lines of the second system. PORISM 11. Given a system of n straight lines in space, and given the first n — 2 straight lines <>f a second system of n straight lines in space : Innumerable straight lines (contained in the surface of a deter- minable hyperboloid of one sheet) can be found, such that if we chose any two of them, and draw a plane through each, we can find two BY MARTIN GARDINER, C.E. 87 points R and S, one in each plane, and through these points draw innumerable pairs of corresponding straight lines (the line through each point being in the plane in which the point lies), such that if we chose any pair of the corresponding lines as the n — 1th and nth straight lines of Hie second system, ive shall render porismatic the problem of the construction of the closed planes ri'gon, whose n planes contain the n lines of the second system, and whose n angular joints rest on the n lines of the first system. PORISM 12. Given the first n — 1 of a system of n straight lines in space, and given also the first n — 1 of a second system ofn straight lines in space : — If a closed planes n'gon have its first n — 1 angular joints on the respective n — 1 lines of the first system, and its n — 1 first planes containing the respective n — 1 lines of the second system ; two straight lines and a point in each can be found, such that if from either of these two points we draw any staight line L through the nth angular joint of the n'gon, and that through the point where the other found line pierces the -D^ plane of the n'gon, we draw any line K in that plane ; then by taking the lines L and K as fixed n"1 lines of the first and second systems, we can deform the rigon so that its n planes will continue to contain the n straight lines of the second system, and its angular joints move on the n straight lines of the first system. Moreover, we can give any position to K , and find innumerable corresponding ansiverable positions for L fall in the surface of a determinable hyperboloid of one sheet.} The following theorems are also obvious consequences from the theory of homographic figures : — THEOREM 7. If there be three distinct closed straight line w'gons having their first angular points in one straight line xx, each ?i'gon of which has its sides passing through n fixed points, and its angular points in n fixed planes ; then will any point in the straight line xx be an answerable position for the first angular point of another such closed w'gon, and any other point in the plane containing xx 88 GEOMETEICAL RESEARCHES, will be an answerable position for the first angular point of a straight line closed 2 w'gon, whose two successive series of n sides will pass in order through the n fixed points, and whose two successive series of angular points will be situated in the n fixed planes. And, reciprocally, if we can form one closed 2 w'gon having its first n sides distinct from its second n sides, &c. THEOREM 8. If there be 4 closed straight line rc'gons, each one of which has its sides passing in order through n . fixed points, and its n angular points resting on n fixed planes, and that the first angular points (all in one plane P) of these closed w'gons are not all in one straight line ; then will any point in the plane P be an answerable position for the first angular point of another such closed tt'gon. NOTE. The principal porisms (concerning plane straight line ^'gons) envolved in this paper contain as particular cases all those of a kindred nature, said to have been comprised in the writings of the ancient Greek Geometers. They contain also as particular cases all those concerning poly- gons which are given in the works of Professors Simsoii, Mulcahy and Chasles. PAPER No. 3. Researches concerning n'gons inscribed in curves of the second degree, by MARTIN GARDINER, C.E. 1. Let S represent any fixed curve of the second degree ; and let o1? og, on represent n fixed points of a series, which, taken in order, may be designated the 1st, 2nd, 3rd, and nm points respectively. BY MARTIN GARDINER, C.E. 89 Suppose fll aa «n+1, ^ 6, i^, ^ c2 .... CR+I, ^ <*a .... r/ &c., to be inscribed ft'gons, the successive sides of each of which pass in order through the n points of the series. If we regard the point ol and its polar as vertex and axis to homologic figures whose homological ratio is — 1, and that we look on the points a^ b^ cv d^ &c., as belonging to one figure, then will «2, b^ cg, d^ &c., be the corresponding points in the other figure. Similarly, if we regard the point o2 and its polar as vertex and axis to homologic figures whose homological ratio is — 1, and that we look on the points «2, b^, cy d^ &c., as belonging to one figure, then will «3, b^, c3, d^ &c., be corresponding points in the other figure. We may thus proceed from the extremities of sides of the n'gons of like subscript numbers to those of higher subscripts until we arrive at the final extremities of the n'gons. And as homologic figures are homographic, and that figures homo- graphic with any figure are homographic with each other, there- fore it is evident the first points a^ b^ cv d^ &c., of the inscribed w'gons, and their final points «n+1, &n+1, CB+I, &c., are correspond- ing points of homographic figures. Moreover, it is evident that the tangents to the cnrve S, at the corresponding points of these homographiclfigures, are corresponding lines of the figures. 2. By indicating tangents to the curve by capital letters of like names and subscripts to the small letters indicating the points of contact, we at once infer the following important theorem : — THEOREM I. If in a curve of the second degree there be three n'gons a1 a2 S° that th* D 90 GEOMETETCAL RESEARCHES, successive sides of each pass in order through n fixed points ov o2, .... on ; then, dl d 2 .... d.^ representing an inscribed rigon ivhose sides pass in like manner through the fixed points., and which we may conceive to be deformed so as to assume the position of all the rigons which can be so inscribed., we will have 3. We know that the first extremities of the inscribed n'gons are corresponding points to their last extremities in a pair of homographic figures. We know also that according as these homographic figures are homologic or not homologic so according- ly will the closing chords of the w'gons all pass through one point or be tangents to a conic T having double contact with S, (in the points which are answerable positions for first extremities of inscribable closed w'gons whose sides pass in order through the given points) and of like or unlike rotatives, (reckoning from the final extremities of the n'gons) just according as the final ex- tremities of the ^'gons are on the same side or on opposite sides of the line of contact of the conies T and S. But, when we can interchange the distinct extremities of one of the inscribable w'gons, we know that the figures are homologic. Hence we have the following theorems : — THEOREM 2. If in a curve of the second degree there can be one inscribed closed 2 rtgon ivhose two successive series of n sides pass in order through n. fixed points, and are not co-incident ; then will any point in the curve be an answerable position for the first extremity of another inscribable closed 2 rigon whose sides will pass in like order through the *& fixed points. And the closing chords of the inscribable rigons the sides of each of which pass in order through the n fixed points will all pass through one point, whose polar cuts the curve in the answerable positions for first extremities of inscribable closed rigons whose sides pass in order through the points. BY MARTIN GARDINER, C.E. 91 THEOREM 3. If in a conic S there can be inscribed one open 2 n'gon whose first n sides and whose second n sides pass in order through n fixed points, then will the closing chords of all the inscribable logons whose sides %)ass in order through the n fixed points be tangents to a conic T having double contact with the given conic S, and they will be of Hike or unlike rotatives in respect to T (reckoning from the final extremities of the logons J just according as the final extremities of the logons are on the same side or on opposite sides of the line of contact. THEOREM 4. If there be a curve of the second degree and a series of n points and that the problem of the inscription of closed n'gons the sides of each of ivhich pass in order through the n points is non-porismatic, there are two and but two answerable positions for the first extremi- ties of such closed rigons. THEOREM 5. If three closed n'gons be inscribable in a curve of the second degree so that the sides of each pass in order through n fixed points of a series of n points ; then will any point in the curve be an answerable position for the first extremity of an inscribable closed utgon whose sides pass in like manner through the n points. This theorem is otherwise evident from the well-known relations of homographic divisions in a conic. It can also be easily deduced from the formula of theorem 1 by supposing a ^ I , and CM , to be co-incident with a^ b^ and c^ respectively. For as = 1, and that -- : -- = 1 it would follow that if dl and d be supposed distinct, the chord f/x d . l should pass through the points of intersection of Aj with B, and 0, 92 GEOMETRICAL RESEARCHES, 4. If o . j be any point in the straight line containing the first extremities al and bl of the two inscribable closed fi'gons, whose sides pass in order through the n fixed points o^ o2, .... on of a series of ^points, it is evident^ a^ ... an al bl b^ ... bn bl a1 is a closed 2 (n -+- 1) 'gon whose two series of n + 1 successive sides pass in order through the n + 1 points o^ oa, .... on, o l Moreover, it is evident (by making a1 the first point of an inscribed 2 (n -f- 1) 'gen) that if the point o 1 be such as to render any point in the curve answerable for the first point of an inscribable closed 2 (n + l)'gon whose sides pass in the prescrib- ed manner through the n -f 1 points o^ o2 on, o.l9 then will 0 , i be situated in the straight line containing a^ and b^ Hence we infer the following theorems : — THEOREM 6. If in a curve of the second degree we inscribe any three distinct tigons^ e2 ...e^, ^ fa . . . . f i+1, g1 g2 .... gn+1, the sides of each of ivhich pass in order through n given points ; then will the three pairs of straight lines GI fn+1, fx e i+1 and GL gn+1, gL en+1 and f, g , ,, g, f , ., cw# eacA other in three points in the straight line ivhich contains the first points of the two inscribable closed logons whose sides pass in order through the n This theorem is otherwise evident, since homographic pencils having a common vertex in a conic, are such that by taking any two pairs of the corresponding radiants and coupling them transversly they will form an involution with the double radiants of the pencils. THEOREM 7. Any point in the straight line containing the first points of the two closed (n — 1) 'gons inscriptible in a curve of the second degree so that the sides of each pass in order through n — 1 fixed points, will be an answerable position for the nth point of the series so as to BY MARTIN GARDINER, C.E. 93 render any point in the curve an answerable position for the first extremity of an inscribable closed 2 n'gon whose two successive series of u sides w ill pass in order through the n points. And no point outside the straight line containing the first extremities of the two closed (n — \) 'gons will possess this property. 5. If we have a curve of the second degree and the first n — 2 points of a series of n points, and that we assume the n — 1th point Off^i anywhere in the straight line containing the first extremities pl and ql of the inscribable closed (w — 2) 'gons whose sides pass in order through the n — 2 given points and that we assume the nth point on co-incident with the pole of the straight line containing the first extremities of the inscribable closed (11 — 1) 'gons whose sides pass in order through the n — 1 points Op o2, .... on_2, °A_1> it is obvious from theorems 6 and 2 that any point in the curve is an answerable position for the first extremity of an inscribable closed w'gon whose sides pass in order through the n points of the series. Moreover, it is evident (by considering p1 or ql as first points of inscribable w'gons) the wth point on lies in the line pl q1 on_r Hence PORISM 1. If any closed n'gon inscribed in a a curve of the second degree have its first n — 2 sides passing in order through n — 2 given points ; then a straight line xx can be found such that if we look on the points in which it is cut by the n — 1th and nth sides of the rigon as fixed i we can deform the n'gon so that its angular points ivill move along the curve and its n sides continue through the n fixed points composed of the n — 2 given ones and the two determined ones. HHJ^ In respect to this porism it is well to remember that the straight line xx cuts the given curve in the answerable positions (real or imaginary) for the first extremities of the inscribable closed (n — 2) 'gons, the sides of each of which pass in order through the n — 2 given points. And when the problem of the inscription of the closed (n — 2) 'gons is porismatic, then o and o must be coincident though otherwise unrestricted in the plane. G 94 GEOMETRICAL RESEARCHES, 6 If we have a conic and any even number of points in one straight line, it is evident the points in which this line cuts the conic, are answerable positions for first angular points of inscrib- able closed n'gons, whose sides pass in order through the n points. Hence from porism 1, we infer the following theorem : — THEOREM 8. Ifn.be any even number, and that in a conic there be inscribed a closed iigon having n — 1 of its sides passing through n — 1 fixed points in one straight line; then will the remaining side pass through a point in the same line, such that if we suppose it fixed, we can de- form the closed rigon so that its angular points will move along the curve, and its n sides continue through the n fixed points. And from theorem 2 or 7, we at once infer the following THEOREM 9. If n be any odd number, and that in a conic there be inscribed any closed 2 rigon such that n — 1 pairs of its opposite sides cut each other in n — 1 points situated in one straight line, then will the re- maining pair of opposite sides cut each other in a point of this same straight line. The particular case in which n = 3 is identical with Pascal's famous theorem concerning an inscribed hexagon. 7. In applying theorem 6 to the finding of the first angular points of the inscribable closed 1'gons, the side of each of which must pass through a given point o^ we immediately perceive that the polar of the point oj cuts the curve in the answerable positions for these angular points. And from this, and porism 1, we have THEOREM 10. If we have a conic and three points, each point of which is the pole of the straight line containing the other two ; then will any point in the conic be an answerable position for the first angular point of an inscribable closed Qgon whose sides pass through the three points taken in any order whatever. BY MARTIN GARDINER, C.E. 95 8. The theorem inverse to that made use of in establishing theorem 8 may be enunciated in the following manner : — THEOREM 11. If two conies, S and T, have double contact (real or imaginary), and that from any four points in the conic S there be drawn tangents to the conic T of like or opposite rotatives just according as the points lie on the same side or on opposite sides of the straight line (always real) containing the points of contact of the conies : then will the anhannonic ratio of the four points whence the tangents are drawn be equal to the anharmonic ratio of the four points in which the tangents touch the conic T, and also to the anharmonic ratio of the other four points in which these tangents again cut the conic S. This theorem is given in a very imperfect form in Chasles' " Geometrie Superieure," and also in Salmon's " Conies" where its discovery is said to be due to Mr. Townsend, of Trinity College, Dublin. The CORRECT THEOREM is now given for the first time. 9. It is evident, from this theorem, and from the preceding portions of the paper, we can form theorems in respect to extended data, analogous to those arrived at. I will give the following one as an instance : — THEOREM 12. If the n angular points of a closed n'gon rest on a curve S of the second degree, and that its first n — 1 sides pass through n — 1 points, or that all or any number of them, not passing through fixed points, are tangents of certain prescribable rotatives to fixed conies having double contacts with S ; then if we deform the Tig on subject to these imposed conditions, the envelope of the n^ side will be a conic having double contact with S, or it will be a determinate point, just according to the possibility of inscribing in S an open 2 (n — 1) 'gon or a closed 2 (n — 1)VOW whose two successive series of n sides are distinct and meet in orderly succession with the n — 1 entities in the manner prescribed for the closed rigons. 96 GEOMETRICAL RESEARCHES, PROBLEM. 10. Given a conic and a series of n points o1, o^ on; to inscribe the closed rc'gons, in the conic, the sides of each of which will pass in order through the points. Now it is evident that if we can find the answerable positions for the first angular points of the closed rc'gons our object will be attained. First method of solution. If n be an odd number, it is obvious from theorem 10 that we can replace the point o and the side of any of the closed w'gons which passes through it by two other points and two chords cutting each other in the curve and passing through the deter- mined points and the extremities of the side which passed through o . Hence we may consider n to be an even number. /sis. — Let al a^ a^ al be one of the inscribed closed ft'gons whose sides pass in order through the n points o > o ... o Looking on the first four sides of this w'gon, let us designate by il the intersection of the lines o1 o^ and o^ o^ ; and let bl be the point in which ^ a^ again cuts the conic. Then, a a a b a being an inscribed 4'gon, it is evident from theorem. 8, that the point r^ in which a^ bl cuts the line ol o2 i^ is known. And, since «3 «4 «5 b± «3 is an inscribed 4'gon, and that o o i are in one line, therefore the point ^ in which the line ag ^ cuts the line °3 °4 *i *s known. Hence we perceive that the inscription of the closed n'gons is reduced to that of the inscription of the closed (n — 2)'gons a^ b «5 .... «w al whose sides pass in order through the n — 2 known points r^ s^ o^ .... on. And thus, step by step, we can reduce the number of sides repeatedly by two until we arrive at a closed 4'gon al b^^ an_1 a^ o^ whose sides pass in order through four known points r , 2 BY MARTIN GARDINER, C.E. 97 Now let i be the intersection of the lines r, s and o |«-1 i«-2 $n-2 n— 1 On ; and let ^^ be the point in which i 9 &n_l again cuts the conic. Since a^ 5 a ^3^-2 ai *s an ^^"bed 4'gon, it follows that the point (see theorem 8) r1 in which a bt cuts r s. •Ji-*- 1 at*-! fi»-2 -511-2 is known. And since an_l ^^ a^ an a^^ is an inscribed 4'gon, the point s, in which a. b , cuts o o is known. Hence, ftt-l 1 4n-l n — 1 n as the points r^^ and s are known, the points a in which the straight, line r^ ^^ cuts the conic are known. Ej§p° I may also remark that porism 1 is prominently evident from this method of solution — for (from the well-known case of theorem 8 in which n = 4) the straight line r s^ is such that *>MT-J 3 -5*1 2 if we assume one of the points on_l5 <>n anywhere therein, we can find a corresponding position for the other one in the same line which will render porismatic the inscription of the closed 4'gon, and .'. also that of the closed w'gon. Second method of solution. Analysis. — Let a a^ . . . . a^ a^ be a closed w'gon inscribed in the desired manner (n being regarded as an even number). Suppose that through «1 we draw the chord «t b.2 parallel to QI a^. Then it is evident (from the well-known particular case of theorem 8 in which n = 4) that the point p^ in which b^ a.3 cuts o, o0 is known. i - And if we suppose the chord b^ b3 parallel to p2 o3, and that we draw &3 «4 to cut p2 o3 in p^ then for like reasons the point j»3 is known. Similarly, if we draw the chord b^ b^ parallel to p »4, and that we draw b a. to cut p3 o , in jj , then will the point pi be known. And proceeding thus, it is evident we at length arrive at the known point pn in which the chord bn «1 cuts the straight line p o . 1 n— 1 n 98 GEOMETRICAL RESEARCHES, The result of these operations is obviously an inscribed closed w'gon al &2 &3 . . . . bn a/l whose first n — 1 sides are parallels to known straight lines QI o^ p2 o^, P$°£ ••" Pn_^ <>n-, and whose last side b a^ passes through the known point p . Now n — 1 being an odd number, we know that the chord bn a1 will be parallel to a fixed determinable direction ; and therefore, as the side is also a transversal through the point pn, we know the point al of its intersection with the conic. Moreover, it is obvious that the point bn will also be an answer- able position for the first angular point of an inscribable closed n'gon fulfilling the conditions. It is also evident that when the point pn is at infinity and indicated by the infinite production of the chords bn a[7 the problem will be porismatic. Third method of solution. Theorem 6 intimates to us Poncelet's elegant method of arriving at the first angular points of the closed w'gons fulfilling the imposed conditions. — Inscribe any three distinct w'gcns ai a2 ••'• VK' &1 52 •••• *n+l' Cl C2 — ' °n+l' the U sideS °f 6ach °f which pass in order through the n given points olt o2, .... o • find *', the point of intersection of the chords a. b and b. a 1 n+l 1 tt-j-i ? find k, the point of intersection of the chords a1 c and ci an4-i ' ^nd ^ ^e P°^n^ °^ intersection of the chords b^ c and GI b._l. Then will the three points *, k, I, be in one straight line which is such that its points of intersection with the conic (real or imaginary as may be) are answerable positions for the first angular points of the inscribable closed ft' This method holds whether n is odd or even, and from the present paper it is obvious it holds in the following more extended problem : — Given a conic S and n entities GI} o^ .... o^ any number of which represent given points, and the rest given conies having double contacts with S ; to inscribe in S the closed BY MARTIN GARDINER, C.E. 99 n'gons, the n sides of each of which will meet with the entities taken in order, in such a manner as that the sides meeting with the entities which are conies will be tangents of prescribed rotatives thereunto. (Remembering that when we fix on the rotative of any particular numbered sides of the w'gons whose first points lie on one side of the line of contact of the entity which they touch, we must have them of opposite rotative when the first points fall on the other side of the line of contact.) Fourth method of solution (for particular case). When the given conic is a circle, the problem of the inscrip- tion of the closed n'gons whose sides pass in order through the n given points o , 02, . . . . o^ can be investigated in the following manner (which is worthy of particular attention as an illustration of the importance of conceiving the methods of rotatives of segments of lines in respect to particular points, and by such means eliminating uncertainty as to which of two straight lines is the answerable one to the object in view.) Method of investi- gation : — Suppose pl p2 • ••• PnPl an inscribed closed rc'gon whose sides pass in order through the n points. Let al a2 .... «n+1 and ^ &2 .... b be two inscribed w'gons, formed at random, the sides of each of which pass in order through the n points. From the properties of similar triangles we have a p a . . p nr 711 * . . 71+1 . , . n"n n-j-1 *\ holding in signs when the rotatives of the involved lines are 100 GEOMETRICAL RESEARCHES, taken in respect to any point in the circumference. And from these we have b° n the rotatives of the involved parts being taken in respect to any point in the circumference. Now since the product of two sides of any plane triangle is equal in magnitude to the product of the diameter of the circum- scribing circle and the perpendicular from their point of inter- section on the third side ; it is evident from the last equation that the perpendiculars from pl on the straight lines al b l and bl a have to each other a known numerical ratio ; and, therefore, all the answerable positions for pl must be included amongst those given by the intersections of the given circle with two known straight lines passing through the intersection of the chords a b and b a . Moreover, since one only of 1 W-f— 1 •*• M— pi these two straight lines gives points jpl fulfilling the equation of conditions when the rotatives of the involved portions are taken in respect to any point in the circumference, it is obvious that the point pl is an intersection of the given circle with one known straight line. By forming another inscribed w'gon cl c2 .... c , x whose sides pass in order through the given points, it is evident the point pl must be in a deter minable straight line through the intersection of the chords al cn+1 and cl a . And hence we infer Poncelet's method of finding the straight line x x containing the answerable positions When an+l and b are co-incident with al and bl then evidently the first side of the last equation is equal unity, and therefore so also the second side. And in this state of the data it is evident that any point in the circumference will be an BY MAETIN GARDINER, C.E. 101 answerable position for pl . Hence we re-arrive at theorem 5, and also at the following : — THEOREM 13. If ^ a2 an ai inscribed in a circle, the sides of each of which pass in order through n fixed points ox, o2, on, and that a o n n when the rotatives of the involved lines are taken in respect to any point in the circumference; then will any point in the circumference be an answerable position for the first angular point of an inscrib- able closed n'gon whose sides pass in order through the n fixed points. 12. When the data is such as to render the problem poris- matic when the number of points is even, and that they are all in one straight line ; then, by supposing the points a^ and bl co-incident with the points in which this line cuts the circle, it is obvious from the last theorem that which is a formula already arrived at by Chasles, on page 465 of his treatise on " Geonietrie Superieure" 13. Again (since the extremities of all the inscribable ^'gons belong to homographic figures) the following theorem can be easily deduced : — THEOREM 14. If there be given a circle and n points in a plane, two straight lines, X and Y can be found, such that if a^ and a be the ex- tremities of any inscribed n'gon ivhose sides pass in order through 102 GEOMETRICAL RESEARCHES, the n points, then will the product of the perpendiculars from al and a on X and Y, respectively , be of constant magnitude. The lines are equidistant from the centre of the circle, and are those corresponding to infinity in the homographic figures to which the extremities of the inscribable n'gons belong. X. ,14{ ]By the well-known process of reciprocation we can at once^form theorems which are the " duals " or "correlatives" of "tJiQ? & investigated, or we can arrive at them by steps correlative to those already used. It is also obvious we can make the solutions of the problems subservient to the solutions of their " duals," or we can arrive at solutions to the dual problems by steps correlative to those used. As an example of the latter mode of proceeding, I will enunciate the dual problem, and give the method of solution correlative to the first. PROBLEM. To exscribe a closed ^'gon to a given curve of the second degree, so that its n successive angular points will be situated in n given straight lines L , L2, .... LM taken in order. Analysis. — Let n be considered an even number ; and let al «2 .... an »j be an exscribed closed M-'gon whose angular points av a.2, .... an rest on LI? L/2, .... Ln respectively. Suppose we draw a straight line I through the points of intersection of LX and L2, and of LS and L4 ; and let i be the point in which this line cuts a2 ay Then if rl be the point in which the other tangent from i cuts al a^ it follows that the straight line RX through ^ and the intersection of LX and L2 is known. And if s1 be the point in which the other tangent from i cuts a^ a^, it follows that the straight line S through 5 and the intersection of Lq and L, is known. o 4 Hence we see that the problem is reduced to the exscribing of BY MARTIN GARDINER, C.E. 103 a closed (n — 2)'gon r s^a^ . . . . a^ TI having its successive angular points on the n — 2 known straight lines B , S1? L&, .... L^. And thus, step by step, we can reduce the problem until we make the solution dependent on that of exscribing a closed 4'gon r 2 s an—\ a r* having its angular points on the four known straight lines R , ST , L , L . f« — Z fn — 2 n — 1 n Now our object is to find out how to form this closed 4'gon. Suppose we draw the straight line which contains the inter- section of the lines R and L^, and the intersection of S* and L _ ; and suppose # to be the point in which this line cuts the Then as the point in which the other tangent from x cuts the side a an must be on each one of two known straight lines (one through the intersection of BA 9 and L , and the other through that of SA 9 and L ) it is known. And therefore the tangent a an through it is known, &c. Moreover it is evident the point x is such that if L and L^ pass through it, the problem of the construction of the closed 4'gon will be porismatic, and therefore also the construction of the exscribed closed ?i'gon. Hence we may announce the following porism, which is the dual of porism 1 already given : — PORISM 2. If a closed n'gon be exscribed to a fixed conic, and have its first n — 2 angular points on n — 2 faed straight lines ; a point x can be found, such that if we draw straight lines from it through the n — 1th and nth angular points of the n'gon, and regard these two lines as fixed, we can deform the n'gon so that its angular points will move along the n fixed straight lines, and its n sides continue tan- gents to the fixed conic. 104 GEOMETRICAL RESEARCHES, 15. It is evident, from the projective properties of figures, that analogous theorems and porisms, to those established con- cerning plane conies, can be established in respect to " spherical conies" It is also obvious that analogous problems concerning spherical conies can be solved by analogous processes. HISTORICAL NOTES : — The problem solved in this paper is famous from having been the chief instrument in unfolding the theory of poles and polars amongst the matchless geometers of France. In 1776, Castillon gave a solution to the particular case in which the conic is a circle and n = 3, which appeared in the " Nemoires of the Academy of Berlin" In 1776, Lagrange indicated a method of arriving at a solu- tion to the particular case considered by Castillon, by means of rather complicated trigonometrical equations. In 1776, the porism pertaining to this case appeared in the " Opera Reliqua " of Professor Simson, of the University of Glasgow. Simson solved the problem in 1731. In 1784, Ottajano and Malfatti (two distinguished Italian Geometers) gave excellent solutions to the more general case in which the curve is a circle and n = any whole number whatever. These solutions were published in the " Memorie della Societa Italiana " of Naples. In 1796, Lhuilier gave a solution to this case, or rather he showed how its solution might be made dependent on the solu- tion of trigonometrical equations. In 1803, the illustrious Carnot (the republican statesman chosen by Napoleon I. to rally the shattered power of the empire against the combined feudalism of Europe) gave a similar solution to this particular case in his work entitled " Geometric de Position" In 1810, Brianchon solved the general problem, in which the curve is any conic, and n any whole number. This solution appeared in the " Journal de P Ecole Poll/technique." In 1817, Poncelet (the celebrated French Engineer) gave an BY MARTIN GARDINER, C.E. 105 elegant and simple method of solution to the general problem in the " Annales des Mathematiques" In 1847, Mr. Townsend (of Trinity College, Dublin) pointed out an easy method of demonstrating the correctness of Poncelet's process by means of anharmonic properties of conies. In 1847, Mr. Gaskin (of Jesus College, Cambridge) furnished a solution which was edited by Professor Davies (of the Royal Military Academy) and published in the " Mechanics' Magazine." This method of solving the general question is evidently nothing more than an extension of the methods of Castillon and Ottajano. Mr. Gaskin has also paid much attention to this problem in an appendix to his work entitled " Solutions to Geometrical Problems; " and he is unquestionably the first geometer who succeeded in arriving at a true conception of the contingent porismatic relations of the data. The inquisitive reader may consult the third volume of " The Mathematician " for a more detailed history, including the labours of Euler, Lexell, Fuss, Gergonne, Servois, Econtre, Rochat, Noble, Wallace, Lowry, Swale, Hearn, &c. PAPEB No. 4. Researches concerning n'gons inscribed in surfaces of the second degree by MARTIN GARDINER, C.E. [Read 17th June, 1863.] 1 . Let S represent a surface of the second degree ; and let o , o2, .... on be a series of n fixed points, designated as first, second, &c., according to the subscript numbers. Let a a^ .... ?*'gons, the sides of each of which pass in order through the n points. 106 GEOMETRICAL RESEARCHES, If we regard the point o]L and its polar plane as vertex and axis to homologic figures whose homological ratio is — 1, and that we look on a^ b , c , d , &c., as belonging to one of these figures, then will a^ &2, cg, d2, &c., be their corresponding points in the other figure. Similarly, by regarding o2 and its polar plane as vertex and axis to homologic figures whose homological ratio is — 1, and looking on a2, &2, c2, d^ &c., as belonging to one of these figures, then will a , b , c3, d , &c., be their corresponding points in the other figure. And it is evident we may thus pro- ceed until we arrive at the final extremities of the inscribed n'gons. But, as homologic figures are homographic, and that figures homographic with any figure are homographic with each other, it is evident the first extremities of the inscribed n'gons belong to a figure which is homographic with a figure to which the final extremities of these ft'gons belong. Moreover, it is evident that the first and last extremities of each %'gon are corresponding points in the homographic figures, and that the tangent planes at the extremities of each w'gon are correspond- ing planes. Hence we may announce the following theorem : — THEOREM 1. If in a surface of the second degree there be inscribed n'gons such that the n successive sides of each pass in order through a series of n fixed points ; then will the first extremities of these argons belong to a figure which is homographic with a figure to which their final extremities belong ; moreover, the extremities of each n'gon will be corresponding points in the homographic figures, and the tangent planes at these extremities will be corresponding planes. 2. The two following theorems are immediate consequences from theorem 1 : — BY MARTIN GARDINER, C.E. 107 THEOREM 2. If in a surface of the second degree n'gons be inscribed whose first extremities are all in one plane, and whose sides pass in order through n fixed points, then will their last extremities be all in one plane. THEOREM 3. If in a surface of the second degree there can be inscribed 3 closed logons whose sides pass in order through n faced points, then will any point in the trace of the plane containing their first extremi- ties be an answerable position for the first extremity of another such inscribable closed n'gon. 3. If in addition to having 3 inscribed closed ?i'gons, whose sides pass in order through the n fixed points, we were to have another such inscribed closed n'gon whose first extremity «1 is not in the trace of the plane containing the first extremities of the other three, then obviously any point yl in the surface is an answerable position for the first extremity of a closed n'gon whose sides pass in order through the n points. — For through x and yl we can conceive a plane whose trace cuts the trace of the plane containing the first extremities of the other three closed w'gons ; and therefore, from theorem 3, it follows that y1 is an answerable position for the first extremity of an inscribable closed Ti'gon. Hence we have — THEOREM 4. If in a surface of the second degree there can be inscribed 4 closed n'gons whose sides pass in order through n fix.ed points, and that the first extremities of these closed rCgons are not all in one plane, then will any point in the surface be an answerable position for the first extremity of another such inscribable closed n'gon. 4. Again (as a consequence from theorem 1) we have the following : — 108 G-EOMETKICAL EESEARCHES, THEOREM 5. If in a surface of the second degree there be inscribed four whose sides pass in order through n fixed points ; then, representing tangent planes by capital letters of like names and subscripts with the small letters indicating the points of contact, we have Bl C«+l' Bl dV BJ rfn+l' Bi And as we can interchange the extremities c^ cn 1? it is evident that , and that = And from HO GEOMETRICAL RESEARCHES, this we learn that the closing chords of all the inscribable open rz-'gons must pass through the straight line zz of intersection of the planes Aj and Br Now, if we conceive a plane through zz and the line GI c.l9 it follows that all the inscribable open w'gons (whose sides pass in order through the points) whose first extremities are in the trace of this plane, will have their final extremities in the same trace. And as the extremities of the %'gons form homographic divisions in the trace, and that two distinct corresponding points Cj and cn are interchangeable in these divisions, therefore it follows that all the closing chords of these inscribable open w'gons will pass through one point v (in the line zz). Moreover, it is evident that the points of contact of the tangents from v to the trace are answerable positions for first extremities of inscribable closed %'gons whose sides pass in order through the n fixed points. Hence, from this and theorem 6, we infer THEOREM 9. If there be a surface of ike second degree and n fixed points, such that one closed 2 rigon can lie inscribed whose two successive series of n sides pass in order through the points and are not coincident ; then will any point in the surface be an answerable position for the first extremity of an inscribable closed 2 n'gon whose sides pass in like manner through the points ; and the problem of the inscription of the closed nfgons whose sides pass in ordef through the n fixed points is partially porismatic. THEOREM 10. If there be a surface of the second degree and a series of n fixed points such that the problem of the inscription of the closed logons whose sides pass in order through the n points is fully porismatic ; then any one of the points of the series being omitted will render partially porismatic the problem of the inscription of the closed (n — \.y gons whose sides pass in the same order through the n — 1 remaining points. And according as the omitted point is inside or outside the surface so will the closed (n — lygons be imaginary or real. H^f0 The trace of the polar plane of the omitted point being the locus of an extremity of a side of the (n — l)'goris. BT MARTIN GARDINER, C.E. HI THEOREM 11. If there be a surface of the second degree and n fixed points, such i der partially porismatic the problem of the inscription of the closed n'gons whose sides pass in order through the n points ; then by adopting the pole of the plane whose trace is the locus of the first extremities of the inscribable closed n'gons, as the n + 1th point of the series, we render fully porismatic the problem of the inscription of the closed (n + I)'gons whose sides pass in order through the series of n + 1 points. THEOREM 12. If there be a surface of the second degree and a series of n fixed points, such that the problem of the inscription of the closed logons whose sides pass in order through the n points is non-porismatic ; then there can be two and not more than two answerable positions (real or imaginary as may be) for the first extremities of the cl> sed ugons. 7. Let us have a surface of the second degree and n points o^ o^ .... o , such as to render non-porismatic the inscription of the closed n'gons whose sides pass in order through these points. And let x x be the straight line containing the first extremities of these closed ?&'gons. If o be any point in the line x x, then evidently the points in which xx pierces the surface are answerable positions for first extremities of inscribable closed 2 (n + l)'gons each of which has its two successive series of n sides passing in order through the n + 1 points o^ o2,....on, on+r Therefore (theorem 9) the problem of the inscription of the closed (n + 1) 'gons whose sides pass in order through the n + 1 points is partially porismatic. Moreover, we know that the line xx is a closing chord of an inscribable (n + l)'gon, and must, therefore, pass through the pole of the plane whose trace is the locus of first extremities of the inscribable closed (n + l)'gons. Hence we have the following theorems : — 1V2 GEOMETRICAL RESEARCHES, THEOREM 13. If there be a surface of the second degree and n points such as to render non-porismatic the problem of the inscription of the closed rigons whose sides pass in order through the points ; then by assum- ing any point in the straight line containing the first extremities of the two inscribable closed rfgons as the n + 1th point of the series, we tvill render partially porismatic the problem of the inscription of the closed (n + 1) 'gons whose sides pass in order through the n + 1 points. THEOREM 14 (porismj. If there be a surface of the second degree and n. fixed points, such as to render non-porismatic the problem of the inscription of the closed rUgons whose sides pass in order through the points ; then, assum- ing any point in the straight line xx containing the first extremities of the two inscribable closed n'gons, as the 11 + 1th point of the series, we can find a position for the n + 2th point of the series in the same straight line which will render fully porismatic the problem of the inscription of the closed (n + 2) 'gons whose sides pass in order through the n + 2 points of the series. 8. And from theorem 14 we at once infer the following theorems : — THEOREM 15. If an open n'gon be inscribed in a surface of the second degree so that its n sides pass in order through n fixed points, and that the problem of the inscription of the closed rfgons whose sides pass in like manner through the points is non-porismatic ; then will the closing chord of the open rigon be in plane with the straight line xx which contains the first extremities of the tiuo inscribable closed n'gons whose sides pass in order through the n fixed points. THEOREM 16. If in a surface of the second degree there be inscribed an open 2 rfgon whose two successive series of n sides pass in order through n fixed povnts ; then will the plane containing its extremities and the first point of its n — 1th side pass through the two answerable positions for the first extremities of the inscribable closed n'gons whose sides pass in order through the n points of the series. BY MARTIN GARDINER, C.E. 113 9. And from theorems 3 and 15 we infer — THEOREM 17. If there be a surface of the second degree and n fixed points, and any odd number k ; and if in the surface there can be inscribed one closed k.n'gon whose k series of successive n sides pass in order through the n fixed points so that no two of the series are co-incident ; then will the first points of the 1st, n + 1th, 2 n + 1th, .... (k — 1) n + 1th, sides lie all in the trace of one plane ; and any point in this trace will be an answerable position for the first extremity of an inscribable closed k.n'gon whose sides pass in order through the n fixed points. 10. The following theorem (of which theorem 9 may be regarded as the particular case in which k = 2) is evident. THEOREM 18. If there be a surface of the second degree and n fixed points, and any even number k ; and if in the surface there can be inscribed a closed k.n'gon whose k series of successive sides pass in order through the n fixed points so that no two of the series of sides are co-incident ; then will the first points of the 1st, n + 1th, 2 n + 1th .... (k — 1) n + 1th sides lie all in the trace of one plane ; and any point in the surface will be an answerable position for the first extremity of another inscribable closed k.n'gon whose sides pass in like manner through the n fixed points; and the problem of the inscription of the closed ^.k.n'gons whose ^.k series of sides pass in order through the n fixed points is partially porismatic. 11. Now let us have a surface S of the second degree and the first n — 2 points o1? og, .... on_0 of a series of n points, such as to render porismatic the problem of the inscription of the closed (n — 2)'gons whose sides pass in order through the n — 2 given points. First, it may be observed that when the inscription of the closed (11 — 2)'gons is fully porismatic, then no distinct fixed positions can be found for the n — 1th and 7ith points which will render porismatic the inscription of the closed tt'gons whose sides pass in order through the n points of the series. But it is GEOMETRICAL RESEARCHES, evident that answerable coincident positions have unlimited space as locus. When the problem of the inscription of the closed (n — 2) 'gons is partially porismatic, we know that the closing chords of inscribed (n — 2) 'gons will pass through the point x which is the pole of the plane X whose trace is the locus of the first extremities of the inscribable closed (n — 2) 'gons. And it is evident that x and X may be real even though the closed (n — 2) 'gons be imaginary. Now if al a2 .... an_l be one of the inscribed (n — 2) 'gons, it is evident that in order to render porismatic the inscription of the closed rc'gons we must have o and o^ in such positions that by drawing a , o , to cut the surface in a , then will a o cut » o n_i n—.i rf n n the surface in a . This can be effected by taking o^_l anywhere in the plane X, and by then taking o anywhere in the polar line of the point o^_ in respect to the trace of X. Hence we have the following theorem : — THEOREM 19 (porismj. Given a surface of the second degree and the first n — 2 points of a series of n points such as to render partially porismatic the problem of the inscription of the closed (n — tycoons whose sides pass in order through the n — 2 given points : a plane X can be found such that by taking the n — 1th point of the series anywhere therein we can find a corresponding straight line in the same plane, any point in which line being made a position for the nih point of the series will render partially porismatic the problem of the inscription of the closed logons whose sides pass in order through the n points of the series. 12. In the investigation of the preceding theorem I have used a theorem arrived at in the researches concerning w'gons inscribed in curves of the second degree, viz. : — " If there be any line of the second degree and 3 points in its plane such that each one has its polar line passing through the other two, then will any point in the curve be an answerable position for the first extremity BY MARTIN GARDINER, C.E. 115 of an inscribable closed 3'gon whose sides pass in any order through the three points ov o9, o3." Now, if we assume the pole of the plane o o0 og as a fourth point, then evidently (see theorem 7) we have THEOREM 20. If there be a surface of the second degree and four points, such that each one is the pole of the plane containing the of her three, then will any point in the surface be an answerable position for the first extremity of an inscribable closed 4<'gon whose sides pass in order through the four points taken in any order. fjj^T The centre of the surface and the three points at infinity indicated by the productions of any system of conjugate diameters, are evidently four points such that each one is the pole of the plane containing the other three. 13. And from theorems 20 and 10 we have THEOREM 21. If there be a surface of the second degree and three points such that the polar plane of each one contains the other two points, then will the problem of the inscription of the closed 3'gons, the sides of which pass in any order through these points, be partially porismatic. fJSp0 These closed 3'gons will be imaginary when the trace of the plane through the three points is imaginary. THEOREM 22. If there be a surface of the second degree and three points such that the polar plane of each one contains the other two points, then any point in the surface 'is an answerable position for the first ex- tremity of an inscribable closed 6'gon whose first three sides and whose second three sides pass through the three points taken in any order. 14. Now if we have a surface of the second degree and any odd number n of points in one straight line, it is evident that the points in which the straight line pierces the surface are answerable lie GEOMETRICAL RESEARCHES, positions for first extremities of inscribable closed 2 w'gons whose first n sides and whose second n sides pass in order through the n points. Hence (see theorem 9) we infer the following : — THEOREM 23. If there be a surface of the second degree and a series n of points (n being an odd number) in one straight line, then will the problem of the inscription of the closed n'gons whose sides pass in order through the n points be partially porismatic ; and any point in tlie surface will be an answerable position for the first extremity of an inscribable closed 2 rig on whose first n sides and whose second n sides pass in order through the n fixed points. Hip33 It is evident that when w=l, the locus of the extremities of the inscribable 1'gons will be the trace of the polar plane of the point through which the sides all pass. 15. The following theorem (of which Pascal's is but a parti- cular case) is an evident consequence. THEOREM 24. If n be an odd number and that in a surface of the second degree there be inscribed a closed 2 iCgon such that all its pairs of opposite sides, with the exception of one pair, cut each other in n — 1 points lying in one straight line, then will this remaining pair of opposite sides cut each other in a point in the same straight line. 16. If n — 2 be an even number, and that the inscription of the closed (n — 2)'gons whose sides pass in order through n — 2 fixed points in one straight line xx is non-porismatic ; then (see theorem 14) by assming any position in xx as a n — 1th point, we can find a corresponding position for a wtb point in the same line so as to render fully porismatic the problem of the inscription of the closed w'gons whose sides pass in order through the n points of the series . BY MARTIN GARDIXER, C.E. 117 This may be formally enunciated, thus : — THEOREM 25. Ifnbe any even number, and that we have a surface of the second degree and n — 1 points in one straight line ;' then a position for a nth point can be found in the same straight line which will render fully porismatic the problem of the inscription of the closed n'gons whose sides pass in order through the n points of the series 17. If there be n fixed points in the plane of a conic, we know- that the problem of the inscription of closed Ti'gons in the conic is either non-porismatic or fully porismatic ; and in the non- porismatic state of the data we can always find a real line containing the two answerable positions for the first extremities of the closed n'gons. Hence we easily arrive at the following theorem. THEOREM 26. If a gauche closed rfgvn inscribed in a surface of the second decree be cut by a plane, and that we conceive the points of its inter- section with the plane to become fixed, then the problem of the inscription of the closed n'gons whose sides pass in order through these n fixed points is partially porismatic or fully porismatic, just according as the problem of the inscription of the closed n'gons in the trace of tJie plane containing the points is non-porismatic or 18. Let the surface S and the n points o1? o2, . . . . on be so related that the problem of the inscription of the closed w'gons whose sides pass in order through the n points is non-porismatic. Let pl and ql be the first extremities of inscribable closed w'gons ; and let xx be the straight line containing these points. Through xx conceive any plane cutting the surface S. £Tow if in the trace of this plane we assume points as first extremities of inscribable open ^'gons whose sides pass through the n fixed points, then will the final extremities of these w'gons be in the same trace. And, as the extremities of each of these ^'gons are corresponding points in homographic divisions in a conic 118 GEOMETRICAL RESEARCHES, such that we cannot interchange them in the divisions, we know that the closing chords of these w'gons are tangents of like or unlike rotatives to a conic k having double contact with the trace in the points where xx pierces it (reckoning from the final ex- tremities of the w'gons) just according as the final extremities of the w'gons are on the same side or on opposite sides of the straight line xx. Similar remarks evidently apply in respect to the traces of all other planes drawn through the line xx. And we know that the conies to which the closing chords are tan- gents belong to a surface having double contact with S in the points where xx pierces it. Again by assuming o , in any particular position in xx, it is evident that the tangent lines from this point to the various conic sections, made in the surface S by planes through xx, are all closing chords to %'gons inscribable in the surface so that the sides of each pass in order through the n fixed points o}, o^ ... o^. And we perceive that we can have any number of closed (n + 1 ) 'gons inscribed in S so that the sides of each pass in order through the n -f 1 points o^ «2, o^, o l; we perceive also that the problem of the inscription of these closed (n + l)'gons is partially porismatic. Hence we learn that the point o is the vertex of a cone of the second degree enveloping the surface which has double contact with S in the points where xx pierces it. This is also evidently true for all other points in xx. There- fore we infer that the envelope of the closing chords of all the inscribable w'gons, whose sides pass in order through the n points, is a surface T of the second degree having double contact with S in the points in which the line xx pierces it. Moreover, it is evident that when we have the rotative of any one of the closing chords (reckoning from the final extremity of the w'gon) in respect to a section of T made by a plane through xx, we can determine on the rotatives of all others, by conceiving the plane to revolve round xx as axis and the chord to be deformed so as to move tangentially to the various conic sections BY MARTIN GARDINER, C.E. 119 of T but not to have either of its extremities pass through the points in which xx pierces the surface. Hence we may announce — THEOREM 27. If there be a surface S of the second degree and a series of n fixed points such as to render non-porismatic the problem of the inscription of the closed logons whose sides pass in order through the n points of the series ; then will the closing chords of the inscribed open logons whose sides pass in order through the fixed points be tangents to a surface of the second degree having double contact with the surface S in the two points which are answerable positions for first extremities of the inscribable closed n'gons whose sides pass m order through the n fixed points. THEOREM 28. If two surfaces of the second degree have double contact ; and that from points in one of the surfaces we draw chords tangent to the other surface in plane with the line xx of contact of the surfaces, and of such roiatives (in respect to the plane sections in this other surface made by the planes through xx containing these tangents) as are indicated by any one of the chords we conceive to revolve with the plane which contains it round the line xx as axis, in such a manner as to be always tangent to the surface but not to have either of its extremities pass through the points where xx pierces the surface; then will the final points of the chords belong to a figure which is homographic with a figure to which the first extremities of these chords belong. THEOREM 29. If two surfaces of the second degree have double contact, and that any point in the line of contact is the vertex of a cone enveloping one of the surfaces, then- will the traces of this cone on the other surface be plane curves. And the poles of the planes containing these traces are situated in the line of contact of the surfaces. THEOREM 30. If there be a surface S of the second degree, and n entities, each entity of which is either a fixed point or a conicoid having double 120 GEOMETRICAL RESEARCHES, contact with S ; and if there be n'gons inscribed in the surface S so that each side of every n'gon will meet with the entity of the series which is of like rank in the series which such side is in the n'gon, and in such a manner as to pass through the entity if it be a point or, if the entity be a conicoid, to be tangent of certain prescribable rota- tive to the trace made on this conicoid by a plane containing the first point of such side and the line of contact of the conicoid with S : then will the extremities of each n'gon be corresponding points of homo- graphic figures. And when we cannot interchange the extremities of any of the n'gons so as to have them still corresponding in the homographic figures, then will the closing chords of the logons be tan- gents of determinate rotatives to the traces made on a certain conicoid having double contact with S by planes containing the final extremi- ties of the n'gons and the line of contact of such conicoid with S. But when we can interchange the distinct extremities of the tigons so that they still remain corresponding points of the hornographic* figures, then will the closing chords of the rigons all pass through one point. 19. It is evident we can extend many of the preceding theorems by a substitution of such entities as those implicated in the theorem just enunciated for some of the entities which we considered as all composed of points. It is also obvious that all these theorems have " duals," which are easy of formation by well known methods — or which can be arrived at by steps correlative to those we have employed in the preceding investigations. — As it would be superfluous to repeat the " duals " of all the theorems, I will content myself by giving that of theorem 20 in order to exhibit a sort of nomenclature which may be found convenient in enunciating the duals of the others. THEOREM 31. If there be a surface of the second degree and four fixed planes such that each plane is the polar of the point common to the other three planes ; then any straight line m any of the planes is an answerable position for an angular joint of a closed planes u'gon whose n planes are tangential to the surface, and whose four succes- sive angular joints lie in the four fixed planes taken in any order whatever. BY MARTIN GARDINER, C.E. 121 20. I will now proceed to indicate methods by which we can graphically find the positions for the first extremities of the closed n'gons inscribable in a surface of the second degree so that the sides of each will pass in order through n given points. And (in doing so) I wo aid have the reader remember that when I speak of a closed w'gon, or of any w'gon, I refer to a n'gon inscribed in the surface whose sides pass in order through the n point of the series ; and wheD I speak of a 2 n'gon, I refer to one inscribed in the surface whose first n sides and whose second n sides pass in order through the n points of the series, and are not coincident in pairs. PROBLEM. Given a surface S of the second degree, and a series of n points Oj, o2, . . . . on ; to inscribe in the surface the closed w'gons the sides of each of which will pass in order through the n fixed points. Analysis of a first method of Solution. Suppose ^ »2 . . . . an a^ to be a closed w'gon such as required, — the sides at a^ a2 ay .... an o^ passing through the respective points ov o2, .... on. Now let us see whether we could reduce the inscription of this closed w'gon to that of another having a less number of sides. If GI, • o2, 03 are in one straight line we know that in this line we can determine a point g such that a^ a]L g will be a straight line. And therefore evidently we can reduce the solution of the problem to the inscription of the closed (n — 2)'gon ^ &4 ag .... an a^ whose sides pass in order through the n — 2 known points If ov oa, o3 be such that each one of them has its polar plane passing through the other two, we know that the point h which is the pole of the plane ol o2 og is such that a^ a^ h is one straight line. And therefore evidently we can reduce the solution of the 122 GEOMETRICAL RESEARCHES, problem to the inscription of a closed (n — 2)'gon whose sides pass in order through the n — 2 known points h, o^ o^ ... o . If o , o2, o , be neither in one straight line nor such that the polar plane of each one passes through the other two, then we can find the straight line xx such that by assuming q any point therein, we can find a corresponding point r, in the same line, such that q a and r a will cut each other in a point p in the surface S. And therefore evidently we can reduce the solution of the problem to the inscription of the closed (n — l)'gon a pa a .... an a whose sides pass in order through the n — 1 known points q, r, o , o&, — o^. So now it is evident we can reduce the solution of the problem of the inscription of the closed w'gons to that of the inscription of closed (n — 2)'gons or to that of the inscription of closed (n — l)'gons. And thus, step by step, we can reduce the problem until we make its solution depend on that of the inscription of closed 3'gons, or 2'gons whose sides are required to pass through known points. When we reduce the problem to the inscription of closed 3'gons whose sides are required to pass through 3 known points, and that these points are in one straight line or that each one of them has its polar plane containing the other two ; then will the problem of the inscription of the closed rc'gons be partially porismatic ; and the locus of the first extremities of the closed w'gons is the trace of the polar plane of the point through which the closing chords of the inscribable open rc'gons all pass. When we reduce the problem to the inscription of closed 2'gons whose sides are required to pass through 2 known points, and that these points are co-incident, then we know that the problem of the inscription of the closed ra'gons is fully porismatic. ll^p0 I need scarcely state that the method of solution just indicated is complete, though it is obvious there are many peculiar states of the data from which we can at once pronounce on the nature of the solution without going through all the indicated operations or processes. BY MARTIN GARDINER, C.E. 123 Analysis of a second method of solution. From theorems 9 and 16 we learn that according as we can inscribe one open 2 w'gon or one closed 2 w'gon, so will the problem of the inscription of the closed w'gons be non-porismatic or partially porismatic. When the data is in the non-porismatic state, it is evident that if we inscribe an open 2 n'gon, and draw the plane which contains its extremities and the first point of its n -f 1th side ; and that we then inscribe another open 2 w'gon whose first extremity is not in this plane ; then will the plane through the extremities and first point of the n + 1th side of this last 2 w'gon cut the other plane in a straight line xx which pierces the surface in the points (real or imaginary as may be) which are the answerable positions for the first extremities of the inscribable closed n'gons. When we can inscribe a closed 2 w'gon ; it is evident that we can inscribe open w'gons, and that the closing chords of these will intersect in a point the trace of whose polar plane is the locus of the answerable positions for the first extremities of the closed w'gons. When the problem is fully porismatic, the fact will be intima- ted to us by our being enabled to inscribe 4 closed w'gons whose first extremities are not all in one plane. ^g^ This method of solution is also complete, and is applica- ble to the following more general problem : — " Given a surface S of the second degree, and n entities in prescribed order, each entity being either a given point, or a conicoid having double contact with S; to inscribe in the surface S closed n'ffons such that each side of each rigon ic ill meet with the entity of the series ivhich is of like rank in the series with such side in the rigon, and in such a manner as to pass through the entity if it be a point, or to be tangent of certain prescribable rotative to the trace made on the conicoid by a plane containing the Jirst point of such side and the line of contact of the conicoid ivith S if it be a conicoid" Third Method of Solution. The following method of finding the first extremities of the 124 GEOMETRICAL RESEARCHES, inscribable closed %'gons is applicable to the more general pro- blem when the data is in the non-porismatic state : — Inscribe three w'gons such that the first extremity of the second ^'gon coincides with the final extremity of the first w'gon, and that the first extremity of the third n'gon coincides with the final extremity of the second w'gon ; draw the plane which con- tains the extremities of these ^'gons, and find its trace on the given surface ; find i the point of intersection of the straight line through the first extremity of the first %'gon and the final extremity of the second %'gon with the tangent line to the trace at the junction of these ^'gons ; find k the point of intersection of the straight line through the first extremity of the second ??.'gon and final extremity of the third w'gon with the tangent line to the trace at the junction of these w'gons : then will the points in which the straight line ik pierces the surface be the answerable positions for the first extremities of the closed w'gons. The proof is obyious from theorem 8, and the properties of the homographic figures in which the extremities of the w'goiis are corresponding points. 21. Various simple solutions can be given to the problem when all the entities are points and that the surface is either spherical, cylindrical or conical. However their exhibition re- quires much more room than can be accorded in this paper, so that I will finish by showing how theorem 5 can be arrived at when the surface is spherical. 22. When the surface is spherical and the entities ov o^....on all points, we can easily derive theorem 5 independently of homologic or homographic considerations. Thus. — Let dt ^••••^n+1 be any variable inscribed w'gon ; and let ^ «2 . . . an+v ^ fta . . . . #n+1, ^ «a .... cw+i> b« thl*ee rc'gons (inscribed at random). From similar triangles we immediately deduce the following relations : — BY MARTIN GAKDIXER, C.E. 2 ' ( From these we at once obtain the relation — (d, M2 (d ,h \2 (c b\* (c , b ,Y \ 1 I/ V »+l «+!/ \ 1 I/ \ «+l n+iy And now since the square of a chord of a sphere is equal to the product of the diameter and the perpendicular let fall from one extremity of the chord on the tangent plane at the other extremity, we perceive (from the last formula) that the following relation (adopting the notation employed in theorem 5) subsists, rfl» Al dn+V A«+l CI> Al Cn+l> A«+l viz. NOTE. Sir William Hamilton, the Astronomer Bo}Tal of Ireland, has given much attention to the problem of this paper. He pub- lished the results of his researches in the Philosophical Magazine for July 1849, and afterwards drew the attention of the Mathe- matical Section of the British Association to the subject. He succeeded in solving only the particular case in which the sur- face is an ellipsoid and the closed n'gon even sided. It seems that his " Quaternion " and other symbolical methods led him to infer that independent of the two positions for the first angular points of the closed w'gons, which may be real or imaginary according to peculiar states of the data, there are also I •426 VARIABLE STARS IN SOUTHERN HEMISPHERE, two necessarily imaginary positions. But it is clearly evident from this paper that such is not the case ; and that his symbo- lical analysis labours under the defect of grasping some extra- neous kindred problem. However, Sir William's method led him to discover theorem 27 as respects the particular state of the data considered : but he does not seem to have observed that the closing chords are all in plane with the line of contact of the surfaces. The solutions which I give to the general problem are ex- tremely simple ; and the numerous new and beautiful theorems unfolded bear testimony to the power of the system of " Geome- tric Superieure " of the modern French School. On the desirability of a systematic search for, and observation of variable stars in the Southern Hemisphere, by ME. JOHN TEBBUTT, JUNR. [Read August 13th. 1862.] THE department of Astronomy relating to variable stars was very little followed up before the middle of the present century. The most complete catalogue that I have yet seen of such objects is that contained in Mr. George F. Chambers' admirable hand- book of Astronomy, published at the close of last year. It comprises 99 stars. Of 22 the dates of discovery are not given, 18 were discovered previous to 1800, and the remaining 59 were discovered during the present century. If we divide the elapsed portion of the present century into periods of ten years, we shall BY JOHN TEBBUTT, JUX. 127 find the following table for the number of variable stars dis- covered in each period : — Number Periods . discovered . 1811—1800 2 1821—1811 0 1831—1821 4 1841—1831 4 1851—1841 18 1861—1851 31 Total 59 It will be observed from the above statement, that within the last twenty, and especially within the last ten years, the zeal of Astronomers in this department of research has been rapidly increasing. Observations of variable stars furnish employment for some of the most distinguished Astronomers of Europe, and as a consequence, we frequently find communications with reference to such objects in the pages of that valuable scientific journal, the Astronomische Naclirichten. Now, considering the additions to our knowledge in this department of Astronomy which have been made during the past twenty years, and are con- stantly being made by Northern observers ; and, also, taking into account the fact that so few stars of high south declination are to be found in the catalogue, the question might well be asked, how is it that so little has been done for the cause in this hemis- phere ? Surely we are not to suppose that our Southern heavens do not furni&h as fertile a field for such discoveries as the Northern Hemisphere does. This scarcity of results does not pro- ceed from any peculiar barrenness of the Southern heavens, but rather from the great want of Southern observers. On an ex- amination of the above npntioned catalogue, it will be seen that out of the 99 stars that it contains, 25 only are situated in the Southern Hemisphere, and of these only two are south of the 23rd parallel of declination, namely : Eta (77) Argus, and Kappa (/c) Corona Australis. I may, however, remark that the star B. A. C. 5656, not contained in that catalogue, is variable, on the authority of the British Association Catalogue. Of Eta (77) Argus I shall speak presently, suffice it just now to say that it is one of 128 VAEIABLE STARS IN SOUTHERN HEMISPHERE, the most, if not the most remarkable, in the whole heavens. It cannot be expected that this -department of the science can receive many accessions at the hands of professional Astronomers in this Hemisphere, they being too much occupied with the advance- ment of standard astronomy to devote much time to the search for variable stars. It is, however, a department in which a great deal might be done by amateurs. Intelligent persons, even if unprovided with instrumental means, might, by means of good eyes, work with considerable advantage, so far as the observa- tion of stars from the first to the fifth magnitude is concerned. Observation of fainter stars will, of course, require telescopic aid, and it sometimes happens that a star, which at its maximum is a brilliant object to the naked eye, dwindles to a telescopic object as it advances towards its minimum. But telescopes of three or four inches aperture would be of great assistance, and these are within the means of many amateurs. I shall now treat of my subject under three different heads, namely : — I. The observation of particular stars known to be variable. II. The examination of stars suspected to be variable. III. A general survey and comparison of all the stars of the Southern Hemisphere. And first, the observation of Stars known to be variable. This course of observation should be pursued in order to deter- mine with increased accuracy the law or progress of increase and decrease of their lustre, together with the exact epochs of their maxima and minima. Mr. Pogson, the present director of the Madras Observatory, who has hitherto taken much interest in this department of Astronomy, forwards periodically to the AstronomiscJie Nachrichten ephemerides of the observed variable stars to assist observers in their researches. We must not infer from the fact of a star being observed to go through its varia- tions in a certain period that it will continue to do so. Some of the stars that were originally thought to undergo fluctuations of brightness in regular periods have been found from continuous observation to go through those variations in irregular intervals of time. Thus I may instance the case of the distinguished star Algol in the constellation Perseus, From careful continuous BY JOHN TEBBUTT, JUN. 129 observation of this star by Argelander, Heis, and Schmidt, its period is found to be shorter now than when it was first dis- covered. The period is found to diminish, not progressively, but with accelerated rapidity. It is impossible to foresee what the final result of this gradual diminution will be. The star may even- tually become of constant lustre, or its period, after having arrived at a certain state, may again lengthen with accelerated rapidity. The law and cycles of its variations have, of course, to be deter- mined from a long course of future observation. There are also great apparent irregularities attending the variations of Omicron (o) Ceti, another remarkable star in the class we are considering. The maximum of this star for 1862, according to Pogson's Ephemeris, occurred on July 3rd, its magnitude then being the second ; it is well known that at its minimum it becomes invisible to the naked eye. It is a star which commends itself to the close attention of Southern observers. Although it is within reach of European observers, still the conditions under which it is seen by most of them are not so favourable as those under which it is seen in this latitude. Few Northern observers have the advantage of clear skies like those of Australia. No observa' tions, perhaps, are more difficult of accurate performance than those attending variable stars, both on account of the many sources of error to which they are liable, and because so much depends on the observer's judgment. The more numerous, then? the observers of a particular star, the greater reliance is to be placed on the general result. Every variable star should, if possible, be observed about the time of its superior meridian transit, as atmospheric causes interfere much with the accuracy of the results. I should be trespassing beyond the limits of a paper were I to enter into a description of all the known variable stars ; it will be sufficient to give a table of those which are ad- vantageously situated for observation in this hemisphere. Such a table will be found appended to this paper. But after having spoken of Algol aud Mira Ceti, I must not forget to say a few words with reference to our remarkable Southern star Eta (77) Argus, which more nearly concerns us in this latitude. This star has, for the last thirty-five years, been known to be variable. 130 VAEIABLE STAES IN SOUTHERN HEMISPHERE, The following extract, from Sir John HersehelPs " Outlines of Astronomy," shows the changes that have been recorded of it previous to 1844 : — " In the time of Halley (1677) it appeared as a star of the fourth magnitude. Lacaille, in 1751, observed it of the second. In the interval from 1811 to 1815 it was again of the fourth ; and again, from 1822 to 1826, of the second. On the 1st of February, 1827, it was noticed by Mr. Burchell to have increased to the first magnitude, and to equal Alpha (a) Crucis. Thence, again, it receded to the second ; and so continued until the end of 1837. All at once, in the beginning of 1838, it suddenly increased in lustre so as to surpass all the stars of the first magnitude except Sirius, Canopus, and Alpha (a) Centauri, which last star it nearly equalled. Thence, it again diminished, but this time not below the first magnitude until April, 1843, when it had again increased so as to surpass Canopus, and nearly equal Sirius in splendour." I have myself watched the variations of this star for some years past with considerable interest. In 1854 it was a very conspicuous object west of the Southern Cross, equilibrating as it were the two bright stars of the Centaur with that con- stellation. The most casual observer of the heavens might now miss the bright object which was then so conspicuous. The following comparisons, made by me in July of that year, may be interesting in conjunction with the present insignificant ap- pearance of the star : — Of Alpha (a) Centauri, Beta (/3) Centauri, Alpha (a) Crucis, and Eta (77) Argus, the first was by far the brightest. Beta (/5) Centauri and Eta (77) Argus were about equal in brilliancy. It was difficult to judge of the comparative brightness of these two, but I considered the latter to be somewhat the brighter. Eta (77) Argus was somewhat brighter than Alpha (a) Crucis. The star in question now appears as one of ftbout the 4| magnitude. A marked diminution of its lustre has taken place since May, 1860. About that time I pointed out to the Rev. W. Scott, the Government Astronomer, the remarkable changes that had taken place in its lustre ; and at his recommen- dation I compared its magnitude with stars within range of the European observatories. I found some difficulty in making this BY JOHN TEBBUTT, JTJN. ]31 comparison, owing to the stars being situated in different parts of the heavens, my observations being made without the assistance of an astrometer. From very careful comparisons, however, I considered its lustre to be equal to that of Beta (/3) Canis Minoris. I, at the same time, compared it with Delta (B) Crucis and Theta (0) Argus, to which I found it equal. This latter estimate may be considered as accurate as can be formed by the unaided judgment, the stars of comparison being moreover in the same part of the heavens. At the close of January, of the present year, I compared Eta (if) Argus with Sigma (2 millions of miles. Now from the known brilliancy of the comet in that portion of its orbit which it traversed while above our horizon from the 1st September to the middle of October it is obvious that it must at the above dates have been far beyond the limits of unassisted vision. In this circumstance, then, it differs from the comet discovered by MM. Tempel and Seeling, which is stated to have 148 ON THE COMET OF SEPTEMBER 1862.— No. 2. been just visible to the naked eye. Again, it will be observed that the motion of our comet, although nearly in the direction of the pole-star, was only a few minutes of arc daily, whereas the other comet moved over the large arc of about seventy degrees in three days. Here, then, is conclusive evidence that the two comets were pursuing totally different orbits. On making known this circumstance to a friend my attention was drawn by him to the Illustrated London News of August 16th, a later date than I myself had received. It was therein stated that a comet had been discovered by Rosa at Rome, on the 25th July, which became distinctly visible to the naked eye on the 3rd August. The notice is accompanied with a rough chart showing the comet's apparent track among the stars from the night of discovery to the 20th August. The chart enables me at once to identify this comet as the one observed here, for on calculating back from the elements, I find our comet occupied precisely the positions indicated in the chart, and must have become visible to the naked eye in the beginning of August. On the night of discovery, the comet was distant one hundred and three millions of miles from the sun, and one hundred and eighteen millions from the earth. The comet discovered by MM. Tempel and Seeling might have been seen in the colony after the first week in July ; but I am not aware that it has been seen by any one. It is usual for Astronomers to designate the comets of any particular year according to the order of their times of perihelion passage. Taking, therefore, into account the fact that the two comets under consideration appeared in Europe almost simultaneously, and our present ignorance respecting the time of the perihelion passage of the one, I am not yet justified in designating our comet as " I. of 1862." I hoped the October mail would bring us a considerable amount of intelligence respecting the late comet, but that hope has not been realised. In the Herald of the 20th ultimo, there is an extract of a letter from Mr. Hind to the Times, which confirms to some extent the results contained in my last paper. He states the comet will be nearest to the sun on the 23rd, (August is implied, though not expressed, in the extract), and distant thirty-two and a half million.-, of miles from the earth BY JOHN TEBBUTT, JUN. 149 on the 30th. He says, further : " the comet will traverse the plane of the ecliptic on the llth September, at a point distant rather more than two millions of miles from the earth's path." Now, it will be remembered that, in my last paper, I stated that the nearest approach of the comet to the earth's path had occurred on the llth September, but that the distance between the two orbits was rather more than half a million of miles. This is about one-fourth of the distance which Mr. Hind gives. It is, however, very probable that Mr. Hind's are only approxi- mate results, based on a few observations taken at the comet's first appearance : it is, of course, usual for astronomers to give rough results at the beginning, to be gradually corrected from future observations. I think, therefore, it will be found, when more mature results reach us from Europe, that the comet actually approached the earth's orbit much nearer than two millions of miles. I speak thus confidently because the elements which I have given satisfy pretty nearly observations extending over six weeks. Further on I will give a comparison of the parabola, with the results derived from observation. From the meagre accounts that have reached us, it appears that the phenomena mentioned in my last paper as attending the comet were also observed in Europe during the month of August. These phenomena are described as most extraordi- nary, so we may expect some interesting information respect- ing them by the next mail from Europe. There is a drawing of the comet in the Illustrated London News of August IGth, from which it will be seen that the comet in its general ap- pearance resembled that of Halley, as figured by Struve on October 8th, 1835. Since the last meeting of the Society I have slightly corrected the elements contained in my former paper. My latest approxi- mation, uncorrected for aberration and parallax, is as follows : — Perihelion passage, 1862, August 23rJ, '1340 Greenwich mean time. Perihelion distance 0.96290 Distance of perihelion from ascending i Deg. min. sec. node, according to the order of the signs J 20" 10 12 Longitude of perihelion on orbit 344 23 48 i Mean equinox of Longitude of ascending node 137 13 36 ) 1st January, 1862. Inclination of orbit 66 9 36 Motion, retrograde. 150 ON THE COMET OF SEPTEMBER 1862.— No. 2. Adopting the above elements, we have the following expres- sions for the comet's heliocentric co-ordinates referred to the plane of the Equator and the two corresponding planes : — Log sin X =[—9-9976609] + log sin (191e23'.96 + v.) Log tan A =[+ 9-0149823] + log tan (191°23'.96 + v.) x = r cos (141° 21 ' 22 + A) cos X y = r sin (141° 21' 22 + A) cos X = x tan (141° 21' 22 + A) z = r sin X where r is the comet's radius vector and v the true anomaly, reckoned according to the order of the signs, or contrary to the direction of the comet's motion. I believe the elements above given, will be found to be a pretty close approximation to the true ones, considering the difficulty attending exact observations of the comet, and the liability to error of the positions of the stars of comparison. In all cases, I have adopted the beso authorities at command for the positions of the stars of comparison. In three instances, I have been enabled to avail myself of corrections deduced from meridian observations made at the Sydney Observatory in 1859 and 18CO. As an instance of the liability to error of the Southern catalogues, I may refer to my observations of the 13th October. In the comparisons of that evening, I employed B. A. C. 5558 a star of the sixth magnitude, and Lacaille 6907 one of the seventh. The latter corresponds with 5789 of the Brisbane Catalogue. My comparisons of the two stars showed that Lacaille 6907 was thirty-five seconds of time west of B. A. C. 5558. Lacaillle's catalogue gives the same result, but Brisbane makes it only sixteen seconds west of B. A. C. 5558. The position of Brisbane 5789 depends on one observation only, and this is the case with many other stars in that catalogue. It appears, therefore, we must be cautious in placing much depen- dence on the star positions of the Southern catalogues. A complete and accurate re-observation of the Southern stars, down to the eighth magnitude inclusive, more especially those between the parallels of 20 and 50 degrees declination, is a work of the highest importance, and without which we cannot expect any great advance in the astronomy of this hemisphere. The best observations of a planet or comet cannot be made available for any accurate purpose till the origins from which its position BY JOHN TEBBUTT, JUX. 151 are measured are known with certainty. For the accomplish- ment of this great work, we must, in a great measure, look to our observatory : a valuable instalment of it has already been presented to the scientific world in the three volumes of observations for 1859, 1860, and 18(51. The following table contains the results of my observations, with the exception of those where the stars of comparison could not be found in the catalogues of the British Association, Brisbane, and Lacaille. The later observations will prove useful in the event of the comet not being more accurately observed elsewhere. The positions are uncorrected for parallax, and referred to the mean equinox and mean equator of January 1st, 18G2. The last two columns of the table exhibit a comparison of some of the places derived from observation with the corresponding positions deduced from the parabola. In order to this comparison, the comet's places have been corrected for parallax. C and O denote respectively the calculated and observed places of the comet, and A a, ATT, the differences of right ascension and north polar distance. The October comparisons indicate the necessity of a further slight correction of the elements, but it would be a waste of time to attempt extreme accuracy, considering the probable amount of the errors of the catalogues, and of the comparison obser- vations themselves. The positions of the loth October, are the latest that can be depended on for a comparison ; the comet's distances from the sun and earth, were then respectively, one hundred and twenty-five and one hundred and forty -six millions of miles. I succeeded in making some observations on the evening of the 21st, but the results, owing to the excessive faint- ness of the comet, are liable to errors of upwards of a minute of arc ; they are, therefore, not sufficiently accurate for a correction of the orbit. Having now given you as much information as I can under the circumstances, I must take leave of the subject. The members of the Society will, doubtless, find much to interest them in the intelligence which will be received respecting the comet by t-ho English mail, now due, and that of December next. 152 ON THE COMET OF SEPTEMBER 1862.— No. 2. i gg £v* OO fr £ « >< 3 p 52; & £< II { I* §2 ? 2 2S 2 £8 00 O O 00 ~ O ^H ++ I I 1+ I I I OO O O OO*-"f«^H II I I + + + ++ «3CO 5N -^ •-H -rt ~F-I -i-«C